9Trigonometric ONLINE PAGE PROOFS. functions 1

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1 9Trigonometric functions 9. Kick off with CAS 9. Trigonometric ratios 9. Circular measure 9. Unit circle definitions 9.5 Smmetr properties 9. Graphs of the sine and cosine functions 9.7 Review

2 9. Kick off with CAS Trigonometric functions Using CAS technolog in radian mode, sketch the following trigonometric functions over. a = sin() b = sin() c = sin() d = 5 sin() e = sin() Using CAS technolog, enter = a sin() into the function entr line and use a slider to change the value of a. When sketching a trigonometric function, what is the effect of changing the value of a in front of sin()? Using CAS technolog in radian mode, sketch the following trigonometric functions over. a = cos() b = cos() c = cos() d = 5 cos() e = cos() 5 Using CAS technolog, enter = a cos() into the function entr line and use a slider to change the value of a. When sketching a trigonometric function, what is the effect of changing the value of a in front of cos()? 7 Using CAS technolog in radian mode, sketch the following trigonometric functions over. a = sin( ) b = cos() c = sin( ) d = cos e = sin 8 Using CAS technolog, enter = cos(n ) into the function entr line and use a slider to change the value of n. 9 When sketching a trigonometric function, what is the effect of changing the value of n in the equation? Please refer to the Resources tab in the Prelims section of our ebookplus for a comprehensive step-b-step guide on how to use our CAS technolog.

3 9. Units & AOS Topic Concept Trigonometric ratios Concept summar Practice questions Interactivit Trigonometric ratios int-577 WorKED EXamPLE Trigonometric ratios The process of calculating all side lengths and all angle magnitudes of a triangle is called solving the triangle. Here we review the use of trigonometr to solve rightangled triangles. right-angled triangles The hpotenuse is the longest side of a right-angled triangle and it lies opposite the 9 angle, the largest angle in the triangle. The other two sides are labelled relative to one of the other angles in the triangle, an eample of which is shown in the diagram. It is likel that the trigonometric ratios of sine, cosine and tangent, possibl together with Pthagoras theorem, will be required to solve a right-angled triangle. sin (θ) = opposite, cos (θ) = adjacent opposite and tan (θ) = hpotenuse hpotenuse adjacent, usuall remembered as SOH, CAH, TOA. Hpotenuse θ Adjacent Opposite These ratios cannot be applied to triangles which do not have a right angle. However, isosceles and equilateral triangles can easil be divided into two right-angled triangles b dropping a perpendicular from the verte between a pair of equal sides to the midpoint of the opposite side. B and large, as first recommended b the French mathematician François Viète in the siteenth centur, decimal notation has been adopted for magnitudes of angles rather than the seagesimal sstem of degrees and minutes; although, even toda we still ma see written, for eample, either 5 or 5. for the magnitude of an angle. Calculate, to decimal places, the value of the pronumeral shown in each diagram. a h b a 8 think a Choose the appropriate trigonometric ratio. WritE a Relative to the angle, the sides marked are the opposite and the hpotenuse. sin ( ) = h 58 maths QuEsT mathematical methods VcE units and

4 Rearrange to make the required side the subject and evaluate, checking the calculator is in degree mode. h = sin ( ) = 5.5 to decimal places b Obtain the hpotenuse length of the lower triangle. In the upper triangle choose the appropriate trigonometric ratio. Rearrange to make the required angle the subject and evaluate. Eact values for trigonometric ratios of, 5, B considering the isosceles right-angled triangle with equal sides of one unit, the trigonometric ratios for 5 can be obtained. Using Pthagoras theorem, the hpotenuse of this triangle will be + = units. 5 The equilateral triangle with the side length of two units can be divided in half to form a right-angled triangle containing the and the angles. This right-angled triangle has a hpotenuse of units and the side divided in half has length = unit. The third side is found using Pthagoras theorem: = units. The eact values for trigonometric ratios of, 5, can be calculated from these triangles using SOH, CAH, TOA. Alternativel, these values can be displaed in a table and committed to memor. θ 5 sin (θ) cos (θ) tan (θ) b From Pthagoras theorem the sides, 8, form a Pthagorean triple, so the hpotenuse is. The opposite and adjacent sides to the angle a are now known. tan (a) = tan (a) =. a = tan (.) = 5.9 to decimal places = = = As a memor aid, notice the sine values in the table are in the order,,. The cosine values reverse this order, while the tangent values are the sine values divided b the cosine values. Topic 9 Trigonometric functions 59

5 For other angles, a calculator, or other technolog, is required. It is essential to set the calculator mode to degree in order to evaluate a trigonometric ratio involving angles in degree measure. WorKED EXamPLE A ladder of length metres leans against a fence. If the ladder is inclined at to the ground, how far eactl is the foot of the ladder from the fence? think Draw a diagram showing the given information. Choose the appropriate trigonometric ratio. Calculate the required length using the eact value for the trigonometric ratio. WritE m m Let the distance of the ladder from the fence be m. Relative to the angle, the sides marked are the adjacent and the hpotenuse. cos ( ) = = cos ( ) = = State the answer. The foot of the ladder is metres from the fence. Deducing one trigonometric ratio from another Given the sine, cosine or tangent value of some unspecified angle, it is possible to obtain the eact value of the other trigonometric ratios of that angle using Pthagoras theorem. One common eample is that given tan (θ) = it is possible to deduce that sin (θ) = and cos (θ) = without evaluating θ. The 5 5 reason for this is that tan (θ) = means that the sides opposite 5 and adjacent to the angle θ in a right-angled triangle would be in the ratio :. θ Labelling these sides and respectivel and using Pthagoras theorem (or recognising the Pthagorean triad ',, 5') leads to the hpotenuse being 5 and hence the ratios sin (θ) = and cos (θ) = are obtained. 5 5 WorKED EXamPLE A line segment AB is inclined at a degrees to the horizontal, where tan (a) =. a Deduce the eact value of sin (a). b Calculate the vertical height of B above the horizontal through A if the length of AB is 5 cm. maths QuEsT mathematical methods VcE units and

6 think a Draw a right-angled triangle with two sides in the given ratio and calculate the third side. WritE a tan (a) = sides opposite and adjacent to angle a are in the ratio :. c area of a triangle a Using Pthagoras theorem: c = + c = State the required trigonometric ratio. sin (a) = b Draw the diagram showing the given information. b Let the vertical height be cm. B Choose the appropriate trigonometric ratio and calculate the required length. A The formula for calculating the area of a right-angled triangle is: a 5 cm sin (a) = 5 = 5 sin (a) Area = (base) (height) For a triangle that is not right-angled, if two sides and the angle included between these two sides are known, it is also possible to calculate the area of the triangle from that given information. cm = 5 as sin (a) = = or State the answer. The vertical height of B above the horizontal through A is cm. Topic 9 TrigonomETric functions

7 WorKED EXamPLE think Consider the triangle ABC shown, where the convention of labelling the sides opposite the angles A, B and C with lower case letters a, b and c respectivel has been adopted in the diagram. In triangle ABC construct the perpendicular height, h, B from B to a point D on AC. As this is not necessaril an isosceles triangle, D is not the midpoint of AC. In the right-angled triangle BCD, c h a sin (C) = h h = a sin (C). a This means the height of triangle ABC is a sin (C) and A D its base is b. b The area of the triangle ABC can now be calculated. Area = (base) (height) b a sin (C) = = ab sin (C) The formula for the area of the triangle ABC, A Δ = ab sin (C), is epressed in terms of two of its sides and the angle included between them. Alternativel, using the height as c sin (A) from the right-angled triangle ABD on the left of the diagram, the area formula becomes A Δ = bc sin (A). It can also be shown that the area is A Δ = ac sin (B). Hence, the area of a triangle is (product of two sides) (sine of the angle included between the two given sides). Area of a triangle: A Δ = ab sin (C) Calculate the eact area of the triangle ABC for which a =, b = 5, c = cm, and A =. Draw a diagram showing the given information. Note: The naming convention for labelling the angles and the sides opposite them with upperand lower-case letters is commonl used. WritE A cm 5 cm B cm C C State the two sides and the angle included between them. State the appropriate area formula and substitute the known values. The given angle A is included between the sides b and c. The area formula is: A Δ = bc sin (A), b = 5, c =, A = A = 5 sin ( ) maths QuEsT mathematical methods VcE units and

8 Evaluate, using the eact value for the trigonometric ratio. A = 5 = = 5 5 State the answer. The area of the triangle is 5 cm. Eercise 9. PRactise Consolidate Appl the most appropriate mathematical processes and tools Trigonometric ratios WE Calculate, to decimal places, the value of the pronumeral shown in each diagram. a h b a 5 Triangle ACB is an isosceles triangle with equal sides CA and CB. If the third side AB has length of cm and the angle CAB is 7, solve this triangle b calculating the length of the equal sides and the magnitudes of the other two angles. WE A ladder of length metres leans against a fence. If the ladder is inclined at 5 to the horizontal ground, how far eactl is the foot of the ladder from the fence? cos ( ) sin (5 ) Evaluate, epressing the answer in eact form with a rational tan (5 ) + tan ( ) denominator. 5 WE A line segment AB is inclined at a degrees to the horizontal, where tan (a) =. a Deduce the eact value of cos (a). b Calculate the run of AB along the horizontal through A if the length of AB is cm. For an acute angle θ, cos (θ) = 5. Calculate the eact values of sin (θ) and tan (θ). 7 7 WE Calculate the eact area of the triangle ABC for which a =, b =, c = cm and C = 5. 8 Horses graze over a triangular area XYZ where Y is km east of X and Z is km from Y on a bearing of N W. Over what area, correct to decimal places, can the horses graze? sin ( ) cos (5 ) 9 a Evaluate, epressing tan ( ) the answer in eact form with a rational denominator. tan (5 ) + cos ( ) b Evaluate, epressing the answer in eact form with a sin ( ) sin (5 ) rational denominator. Topic 9 Trigonometric functions

9 a For an acute angle θ, obtain the following trigonometric ratios without evaluating θ. i Given tan (θ) =, form the eact value of sin (θ). ii Given cos (θ) = 5, form the eact value of tan (θ). iii Given sin (θ) = 5, form the eact value of cos (θ). b A right-angled triangle contains an angle θ where sin (θ) =. If the longest side 5 of the triangle is cm, calculate the eact length of the shortest side. In order to check the electricit suppl, a technician uses a ladder to reach the top of an electricit pole. The ladder reaches 5 metres up the pole and its inclination to the horizontal ground is 5. a Calculate the length of the ladder to decimal places. b If the foot of the ladder is moved.5 metres closer to the pole, calculate its new inclination to the ground and the new vertical height it reaches up the electricit pole, both to decimal place. a An isosceles triangle ABC has sides BC and AC of equal length 5 cm. If the angle enclosed between the equal sides is, calculate: i the area of the triangle to decimal places ii the length of the third side AB to decimal places. b An equilateral triangle has a vertical height of cm. Calculate the eact perimeter and area of the triangle. c Calculate the area of the triangle ABC if, using the naming convention, a =, b = cm and C =. The two legs of a builder s ladder are of length metres. The ladder is placed on horizontal ground with the distance between its two feet of.75 metres. Calculate the magnitude of the angle between the legs of the ladder. Triangle ABC has angles such that CAB = and ABC = 5. The perpendicular distance from C to AB is 8 cm. Calculate the eact lengths of each of its sides. 5 A cube of edge length a units rests on a horizontal table. Calculate: a the length of the diagonal of the cube in terms of a b the inclination of the diagonal to the horizontal, to decimal places. AB is a diameter of a circle and C is a point on the circumference of the circle such that CBA = 8 and CB =.8 cm. a Calculate the length of the radius of the circle to decimal places. b Calculate the shortest distance of CB from the centre of the circle, to decimal place. Maths Quest MATHEMATICAL METHODS VCE Units and

10 7 In the diagram, angles ABC and ACD are right angles and DE is parallel to CA. Angle BAC is a degrees and the length measures of AC and BD are m and n respectivel. A E a Master 9. a Show that n = m sin (a), where sin (a) is the notation for the square of sin (a). b If angle EBA is and CD has length measure of, calculate the values of a, m and n. 8 A lookout tower is metres in height. From the top of this tower, the angle of depression of the top of a second tower stood on the same level ground is ; from the bottom of the lookout tower, the angle of elevation to the top of the second tower is 5. Calculate the height of the second tower and its horizontal distance from the lookout tower, epressing both measurements to decimal place. 9 The distances shown on a map are called projections. The give the horizontal distances between two places without taking into consideration the slope of the line connecting the two places. If a map gives the projection as 5 km between two points which actuall lie on a slope of, what is the true distance between the points? A beam of length metres acts as one of the supports for a new fence. This beam is inclined at an angle of 5 to the horizontal. Calculate the eact horizontal distance of its foot from the fence. Circular measure B Measurements of angles up to now have been given in degree measure. An alternative to degree measure is radian measure. This alternative can be more efficient for certain calculations that involve circles, and it is essential for the stud of trigonometric functions. n D m C Units & AOS Topic Concept Circular measure Concept summar Practice questions Definition of radian measure Radian measure is defined in relation to the length of an arc of a circle. An arc is a part of the circumference of a circle. One radian is the measure of the angle subtended at the centre of a circle b an arc equal in length to the radius of the circle. Topic 9 Trigonometric functions 5

11 In particular, an arc of unit subtends an angle of one radian at the centre of a unit circle, a circle with radius unit and, conventionall, a centre at the origin. units unit unit radian unit unit radians unit Doubling the arc length to units doubles the angle to radians. This illustrates the direct proportionalit between the arc length and the angle in a circle of fied radius. The diagram suggests an angle of one radian will be a little less than since the sector containing the angle has one edge curved and therefore is not a true equilateral triangle. The degree equivalent for radian can be found b considering the angle subtended b an arc which is half the circumference. The circumference of a circle is given b r so the circumference of a unit circle is. In a unit circle, an arc of units subtends an angle of radians at the centre. But we know this units angle to be 8. This gives the relationship between radian and 8 degree measure. radians = 8 (, ) Hence, radian equals 8, which is approimatel 57. ; equals radians, which is approimatel 8.75 radians. From these relationships it is possible to convert from radians to degrees and vice versa. To convert radians to degrees, multipl b 8. To convert degrees to radians, multipl b 8. Radians are often epressed in terms of, perhaps not surprisingl, since a radian is a circular measure and is so closel related to the circle. Notation radian can be written as c, where c stands for circular measure. However, linking radian measure with the length of an arc, a real number, has such importance that the smbol c is usuall omitted. Instead, the onus is on degree measure to alwas include the degree sign in order not to be mistaken for radian measure. Maths Quest MATHEMATICAL METHODS VCE Units and

12 WorKED EXamPLE 5 a Convert to radian measure. b Convert c to degree measure. c Convert to degree measure and hence state the value of sin think WritE a Convert degrees to radians. a To convert degrees to radians, multipl b 8. = 8 = 8 b Convert radians to degrees. Note: The degree sign must be used. Etended angle measure B continuing to rotate around the circumference of the unit circle, larger angles are formed from arcs which are multiples of the circumference. For instance, an angle of radians is formed from an arc of length units created b one and a half revolutions of the unit circle: = +. This angle, in degrees, equals + 8 = 5 and its endpoint on the circumference of the circle is in the same position as that of 8 or c ; this is the case with an other angle which is a multiple of added to c. What is important here is that this process can continue indefinitel so that an real number, the arc length, can be associated with a radian measure. The real number line can be wrapped around the circumference of the unit circle so that the real number θ corresponds to the angle θ in radian measure. B convention, the positive reals wrap around the circumference anticlockwise while the negative reals wrap clockwise, with. = b To convert radians to degrees, multipl b 8. c = 8 = 8 = c Convert radians to degrees. c = 8 = 5 Calculate the trigonometric value. sin = sin (5 ) = Topic 9 TrigonomETric functions 7

13 WorKED EXamPLE the number zero placed at the point (, ) on the unit circle. The wrapping of the real number line around the circumference results in man numbers being placed in the same position on the unit circle s circumference. 5 a Convert c to degree measure. (, ) Positive real number Negative real number b Draw a unit circle diagram to show the position the real number is mapped to when the real number line is wrapped around the circumference of the unit circle. think WritE a Convert radians to degrees. a c = 8 As the can t be cancelled, a calculator is used to evaluate. c 7.9 b State how the wrapping of the number line is made. Draw the unit circle diagram and mark the position of the number. b The number zero is placed at the point (, ) and the negative number line is wrapped clockwise around the circumference of the unit circle through an angle of 7.9 so that the number is its endpoint. c ~ 7.9 (, ) using radians in calculations From the definition of a radian, for an circle of radius r, an angle of c is subtended at the centre of the circle b an arc of length r. So, if the angle at the centre of this circle is θ c, then the length of the arc subtending this angle must be θ r. 8 maths QuEsT mathematical methods VcE units and

14 WorKED EXamPLE 7 This gives a formula for calculating the length of an arc. l = rθ In the formula l is the arc length and θ is the angle, in radians, subtended b the arc at the centre of the circle of radius r. An angles given in degree measure will need to be converted to radian measure to use this arc length formula. Some calculations ma require recall of the geometr properties of the angles in a circle, such as the angle at the centre of a circle is twice the angle at the circumference subtended b the same arc. major and minor arcs For a minor arc, θ < and for a Major arc AB major arc θ >, with the sum of the minor and major arc angles totalling if the major and minor arcs have θ their endpoints on the same chord. To calculate the length of the major θ arc, the refle angle with θ > should be used in the arc length A B A B formula. Alternativel, the sum of Minor arc AB the minor and major arc lengths gives the circumference of the circle, so the length of the major arc could be calculated as the difference between the lengths of the circumference and the minor arc. Trigonometric ratios of angles epressed in radians Problems in trigonometr ma be encountered where angles are given in radian mode and their sine, cosine or tangent value is required to solve the problem. A calculator, or other technolog, can be set on radian or rad mode and the required trigonometric ratio evaluated directl without the need to convert the angle to degrees. Care must be taken to ensure the calculator is set to the appropriate degree or radian mode to match the measure in which the angle is epressed. Care is also needed with written presentation: if the angle is measured in degrees, the degree smbol must be given; if there is no degree sign then it is assumed the measurement is in radians. a An arc subtends an angle of 5 at the centre of a circle of radius cm. Calculate the length of the arc to decimal places. b Calculate, in degrees, the magnitude of the angle that an arc of length cm subtends at the centre of a circle of radius 5 cm. think a The angle is given in degrees so convert it to radian measure. WritE a θ = 5 θ c = 5 = 5 8 Topic 9 TrigonomETric functions 9

15 Calculate the arc length. b Calculate the angle at the centre of the circle subtended b the arc. Convert the angle from radians to degrees. Eercise 9. PRactise Circular measure b l = rθ, r =, θ = 5 l = 5 = The arc length is 9.77 cm (to decimal places). l = rθ, r = 5, l = 5θ = θ = 5 = The angle is radians. In degree measure: θ = 8 = The magnitude of the angle is. WE5 a Convert to radian measure. b Convert c to degree measure. c Convert to degree measure and hence state the value of tan. Epress 5 ' in radian measure, correct to decimal places. WE a Convert.8 c to degree measure. b Draw a unit circle diagram to show the position the real number.8 is mapped to when the real number line is wrapped around the circumference of the unit circle. The real number line is wrapped around the circumference of the unit circle. Give two positive and two negative real numbers which lie in the same position as the following numbers. a b 5 WE7 a An arc subtends an angle of 75 at the centre of a circle of radius 8 cm. Calculate the length of the arc, to decimal places. b Calculate, in degrees, the magnitude of the angle that an arc of length cm subtends at the centre of a circle of radius cm. 7 Maths Quest MATHEMATICAL METHODS VCE Units and

16 he number Consolidate Appl the most appropriate mathematical processes and tools Evaluate the following to decimal places. a tan (.) b tan (. ) 7 a Cop, complete and learn the following table b heart. Degrees 5 Radians b Cop, complete and learn the following table b heart. Degrees Radians 8 Convert the following to degrees. a c b c 5 d e Convert the following to radian measure. c 5 f 9 a b 5 c 5 d e 5 f 7 a Epress in radian measure, to decimal places. i ii 5 iii 5. b Epress in degree measure to decimal places. i c ii. c Rewrite.5 c, 5, c with the magnitudes of the angles ordered from 7 smallest to largest. a For each of the following, draw a unit circle diagram to show the position of the angle and the arc which subtends the angle. i An angle of radians ii An angle of radians iii An angle of b For each of the following, draw a unit circle diagram with the real number line wrapped around its circumference to show the position of the number and the associated angle subtended at the centre of the circle. i The number ii The number iii The number 7 Calculate the eact lengths of the following arcs. a The arc which subtends an angle of 5 at the centre of a circle of radius cm b The arc which subtends an angle of c at the circumference of a circle of radius cm 9 c The major arc with endpoints on a chord which subtends an angle of at the centre of a circle of radius cm a A ball on the end of a rope of length.5 metres swings through an arc of 75 cm. Through what angle, to the nearest tenth of a degree, does the ball swing? Topic 9 Trigonometric functions 7

17 MastEr b A fied point on the rim of a wheel of radius metres rolls along horizontal ground at a speed of m/s. After 5 seconds, calculate the angle the point has rotated through and epress the answer in both degrees and radians. c An analogue wristwatch has a minute hand of length mm. Calculate, to decimal places, the arc length the minute hand traverses between 9.5 am and 9.5 am. d An arc of length cm subtends an angle of.5 at the circumference of a circle. Calculate the area of the circle correct to decimal place. a Calculate the following to decimal places. i tan ( c ) ii cos iii sin (. ) 7 b Complete the following table with eact values. θ sin (θ) cos (θ) tan (θ) 5 a Calculate the area of triangle ABC in which a =, c =, B =. b Calculate the eact value of in the following diagram. a The Western Australian towns of Broome(B) and Karonie(K) N both lie on approimatel the same longitude. Broome is approimatel 9 km due north of Karonie (the distance being measured along the meridian). When the sun is O directl over Karonie, it is. south of Broome. Use this information to estimate the radius of the Earth. B This method dates back to Eratosthenes in 5 bc, although K S he certainl didn t use these Australian towns to calculate his results. b A ship sailing due east along the equator from the Galapagos Islands to Ecuador travels a distance of nautical miles. If the ship s longitude changes from 9 W to 8 W during this journe, estimate the radius of the Earth, given that nautical mile is approimatel.85 km. c Taking the radius of the earth as 7 km, calculate the distance, to the nearest kilometre, along the meridian between place A, located S, E, and place B, located N, E. 7 Convert 5 to radian measure using the mth TRIG function on a CAS calculator. 8 Convert 5 radians to degree measure using the mth TRIG function, epressing the answer both as an eact value and as a value to decimal places. 7 maths QuEsT mathematical methods VcE units and

18 9. Units & AOS Topic Concept Unit circle definitions Concept summar Practice questions Interactivit The unit circle int-58 Unit circle definitions With the introduction of radian measure, we encountered positive and negative angles of an size and then associated them with the wrapping of the real number line around the circumference of a unit circle. Before appling this wrapping to define the sine, cosine and tangent functions, we first consider the conventions for angle rotations and the positions of the endpoints of these rotations. Trigonometric points The unit circle has centre (, ) and radius unit. Its Cartesian equation is + Quadrant Quadrant =. (, ) The coordinate aes divide the Cartesian plane + = into four quadrants. The points on the circle which lie on the boundaries between the quadrants are the endpoints of the horizontal and vertical (, ) O A (, ) diameters. These boundar points have coordinates (, ), (, ) on the horizontal ais and (, ), (, ) (, ) on the vertical ais. Quadrant Quadrant A rotation starts with an initial ra OA, where A is the point (, ) and O (, ). Angles are created b rotating the initial ra anticlockwise for positive angles and clockwise for negative angles. If the point on the circumference the ra reaches after a rotation of θ is P, then AOP = θ and P is called the trigonometric point [θ]. The angle of rotation θ ma be measured in radian or degree measure. In radian measure, the value of θ corresponds to the length of the arc AP of the unit circle the rotation cuts off. 8 O 9, A (, ) P[θ] or P(, ) O θ A (, ) [] O [ ] [], [] A (, ) 7 The point P[θ] has Cartesian coordinates (, ) where: >, > if P is in quadrant, < θ < <, > if P is in quadrant, < θ < ] ] Topic 9 Trigonometric functions 7

19 WorKED EXamPLE 8 think <, < if P is in quadrant, < θ < >, < if P is in quadrant, < θ < Continued rotation, anticlockwise or clockwise, can be used to form other values for θ greater than, or values less than, respectivel. No trigonometric point has a unique θ value. The angle θ is said to lie in the quadrant in which its endpoint P lies. a Give a trigonometric value, using radian measure, of the point P on the unit circle which lies on the boundar between the quadrants and. b Identif the quadrants the following angles would lie in: 5,, c, c. c Give two other trigonometric points, Q and R, one with a negative angle and one with a positive angle respectivel, which would have the same position as the point P[5 ]. a State the Cartesian coordinates of the required point. Give a trigonometric value of this point. Note: Other values are possible. b Eplain how the quadrant is determined. Identif the quadrant the endpoint of the rotation would lie in for each of the given angles. WritE a The point (, ) lies on the boundar of quadrants and. An anticlockwise rotation of 8 or c from the point (, ) would have its endpoint at (, ). The point P has the trigonometric value []. b For positive angles, rotate anticlockwise from (, ); for negative angles rotate clockwise from (, ). The position of the endpoint of the rotation determines the quadrant. Rotating anticlockwise 5 from (, ) ends in quadrant ; rotating anticlockwise from (, ) through would end in quadrant ; rotating anticlockwise from (, ) b of would end in quadrant ; rotating clockwise from (, ) b would end in quadrant. [] ] ] [5 ] [ ] (, ) ] ] State the answer. The angle 5 lies in quadrant, in quadrant, c in quadrant, and c in quadrant. 7 maths QuEsT mathematical methods VcE units and

20 c Identif a possible trigonometric point Q. Identif a possible trigonometric point R. c A rotation of in the clockwise direction from (, ) would end in the same position as P[5 ]. Therefore the trigonometric point could be Q[ ]. A full anticlockwise revolution of plus another anticlockwise rotation of 5 would end in the same position as P[5 ]. Therefore the trigonometric point could be R[ ]. Unit circle definitions of the sine and cosine functions Consider the unit circle and trigonometric point P[θ] with Cartesian coordinates (, ) on its circumference. In the triangle ONP, NOP = θ = AOP, ON = and NP =. As the triangle ONP is right-angled, cos (θ) = = and sin (θ) = =. This enables the following definitions to be given. For a rotation from the point (, ) of an angle θ with endpoint P[θ] on the unit circle: cos (θ) is the -coordinate of the trigonometric point P[θ] sin (θ) is the -coordinate of the trigonometric point P[θ] The importance of these definitions is that the enable sine and cosine functions to be defined for an real number θ. With θ measured in radians, the trigonometric point [θ] also marks the position the real number θ is mapped to when the number line is wrapped around the circumference of the unit circle, with zero placed at the point (, ). This relationship enables the sine or cosine of a real number θ to be evaluated as the sine or cosine of the angle of rotation of θ radians in a unit circle: sin (θ) = sin (θ c ) and cos (θ) = cos (θ c ). The sine and cosine functions are f : R R, f(θ) = sin (θ) and f : R R, f(θ) = cos (θ). The are trigonometric functions, also referred to as circular functions. The use of parentheses in writing sin (θ) or cos (θ) emphasises their functionalit. The mapping has a man-to-one correspondence as man values of θ are mapped to the one trigonometric point. The functions have a period of since rotations of θ and of + θ have the same endpoint on the circumference of the unit circle. The cosine and sine values repeat after each complete revolution around the unit circle. For f(θ) = sin (θ), the image of a number such as is f() = sin () = sin ( c ). This is evaluated as the -coordinate of the trigonometric point [] on the unit circle. The values of a function for which f(t) = cos (t), where t is a real number, can be evaluated through the relation cos (t) = cos (t c ) as t will be mapped to the trigonometric point [t] on the unit circle. θ N P[θ] or P(, ) A(, ) Topic 9 Trigonometric functions 75

21 WorKED EXamPLE 9 think The sine and cosine functions are periodic functions which have applications in contets which ma have nothing to do with angles, as we shall stud in later chapters. a Calculate the Cartesian coordinates of the trigonometric point P show the position of this point on a unit circle diagram. b Illustrate cos ( ) and sin () on a unit circle diagram. c Use the Cartesian coordinates of the trigonometric point [] to obtain the values of sin () and cos (). d If f(θ) = cos (θ), evaluate f(). WritE a State the value of θ. a P This is the trigonometric point with θ =. Calculate the eact Cartesian coordinates. Note: The eact values for sine and cosine of c, or, need to be known. Show the position of the given point on a unit circle diagram. The Cartesian coordinates are: = cos (θ) = sin (θ) = cos c = cos = cos ( ) = = sin c = sin = sin ( ) = Therefore P has coordinates P or P, of the unit circle. (, ),. and lies in quadrant on the circumference P [ ] or P (.5, ) (, ).5 (, ) (, ) 7 maths QuEsT mathematical methods VcE units and

22 b Identif the trigonometric point and which of its Cartesian coordinates gives the first value. State the quadrant in which the trigonometric point lies. Identif the trigonometric point and which of its Cartesian coordinates gives the second value. State the quadrant in which the trigonometric point lies. 5 Draw a unit circle showing the two trigonometric points and construct the line segments which illustrate the - and -coordinates of each point. b cos ( ): The value of cos ( ) is given b the -coordinate of the trigonometric point [ ]. The trigonometric point [ ] lies in quadrant. sin (): The value of sin () is given b the -coordinate of the trigonometric point []. As.57 < <., the trigonometric point [] lies in quadrant. For each of the points on the unit circle diagram, the horizontal line segment gives the -coordinate and the vertical line segment gives the -coordinate. sin () (, ) (, ) [] cos ( ) (, ) (, ) [ ] Label the line segments which The value of cos ( ) is the length measure of represent the appropriate the horizontal line segment. coordinate for each point. The value of sin () is the length measure of the vertical line segment. The line segments illustrating these values are highlighted in orange on the diagram. c State the Cartesian coordinates c An anticlockwise rotation of from (, ) gives of the given point. the endpoint (, ). The trigonometric point [] is the Cartesian point (, ). State the required values. The point (, ) has =, =. Since = cos (θ), cos () = = Since = sin (θ), sin () = = Topic 9 Trigonometric functions 77

23 d Substitute the given value in the function rule. Identif the trigonometric point and state its Cartesian coordinates. Evaluate the required value of the function. d f(θ) = cos (θ) f() = cos () Unit circle definition of the tangent function Consider again the unit circle with centre O(, ) containing the points A(, ) and the trigonometric point P[θ] on its circumference. A tangent line to the circle is drawn at point A. The radius OP is etended to intersect the tangent line at point T. For an point P[θ] on the unit circle, tan (θ) is defined as the length of the intercept AT that the etended ra OP cuts off on the tangent drawn to the unit circle at the point A(, ). The trigonometric point [] has Cartesian coordinates (, ). The value of cos () is given b the -coordinate of the point (, ). cos () = f() = A (, ) Intercepts that lie above the -ais give positive tangent values; intercepts that lie below the -ais give negative tangent values. Unlike the sine and cosine functions, there are values of θ for which tan (θ) is undefined. These occur when OP is vertical and therefore parallel to the tangent line through A(, ); these two vertical lines cannot intersect no matter how far OP is etended. The values of tan and tan, for instance, are not defined. The value of tan (θ) can be calculated from the coordinates (, ) of the point P[θ], provided the -coordinate is not zero. Using the ratio of sides of the similar triangles ONP and OAT: AT OA = NP ON tan (θ) = Hence: tan (θ) =,, where (, ) are the coordinates of the trigonometric point P[θ]. Since = cos (θ), = sin (θ), this can be epressed as the relationship: θ N P T + tangent tan (θ) sin (θ) tan (θ) = cos (θ) 78 Maths Quest MATHEMATICAL METHODS VCE Units and

24 WorKED EXamPLE think Domains and ranges of the trigonometric functions The domain and range of the unit circle require and so cos (θ) and sin (θ). Since θ can be an real number, this means that: the function f where f is either sine or cosine has domain R and range [, ]. Unlike the sine and cosine functions, the domain of the tangent function is not the set of real numbers R since tan (θ) is not defined for an value of θ which is an odd multiple of. Ecluding these values, intercepts of an size ma be cut off on the tangent line so tan (θ) R. This means that the function f where f is tangent has domain R \ ±, ±, and range R. The domain of the tangent function can be written as R \ (n + ), n Z and the tangent function as the mapping f : R \ (n + ), n Z R, f(θ) = tan (θ). a Illustrate tan ( ) on a unit circle diagram and use a calculator to evaluate tan ( ) to decimal places. b Use the Cartesian coordinates of the trigonometric point P[] to obtain the value of tan (). a State the quadrant in which the angle lies. Draw the unit circle with the tangent at the point A(, ). Note: The tangent line is alwas drawn at this point (, ). Etend PO until it reaches the tangent line. State whether the required value is positive, zero or negative. WritE a lies in the second quadrant. [ (, ) ] P (, ) (, ) A(, ) T tan ( ) Tangent Let T be the point where the etended radius PO intersects the tangent drawn at A. The intercept AT is tan ( ). The intercept lies below the -ais, which shows that tan ( ) is negative. 5 Calculate the required value. The value of tan ( ) =.9, correct to decimal places. b Identif the trigonometric point and state its Cartesian coordinates. b The trigonometric point P[] is the endpoint of a rotation of c or 8. It is the Cartesian point P(, ). Topic 9 TrigonomETric functions 79

25 Calculate the required value. The point (, ) has =, =. Since tan (θ) =, Check the answer using the unit circle diagram. Eercise 9. PRactise Unit circle definitions tan () = = Check: PO is horizontal and runs along the -ais. Etending PO, it intersects the tangent at the point A. This means the intercept is, which means tan () =. () (, ) (, ) (, ) A (, ) WE8 a Give a trigonometric value, using radian measure, of the point P on the unit circle which lies on the boundar between the quadrants and. b Identif the quadrants the following angles would lie in:,, c, c. c Give two other trigonometric points, Q with a negative angle and R with a positive angle, which would have the same position as the point P[ ]. Using a positive and a negative radian measure, state trigonometric values of the point on the unit circle which lies on the boundar between quadrants and. WE9 a Calculate the Cartesian coordinates of the trigonometric point P and show the position of this point on a unit circle diagram. b Illustrate sin (5 ) and cos () on a unit circle diagram. c Use the Cartesian coordinates of the trigonometric point values of sin and cos. to obtain the d If f(θ) = sin (θ), evaluate f(). Identif the quadrants where: a sin (θ) is alwas positive b cos (θ) is alwas positive. 5 WE a Illustrate tan ( ) on a unit circle diagram and use a calculator to evaluate tan ( ) to decimal places. T Tangent 8 Maths Quest MATHEMATICAL METHODS VCE Units and

26 Consolidate Appl the most appropriate mathematical processes and tools b Use the Cartesian coordinates of the trigonometric point P[] to obtain the value of tan (). 5 Consider tan ( ), tan, tan ( 9 ), tan a Which elements in the set are not defined? b Which elements have negative values?, tan (78 ). 7 Identif the quadrant in which each of the following lies. a 585 b c 8 d 7 8 Consider O, the centre of the unit circle, and the trigonometric points P and Q on its circumference. 5 a Sketch the unit circle showing these points. b How man radians are contained in the angle QOP? c Epress each of the trigonometric points P and Q with a negative θ value. d Epress each of the trigonometric points P and Q with a larger positive value for θ than the given values P and Q 5. 9 a Calculate the Cartesian coordinates of the trigonometric point P. b Epress the Cartesian point P(, ) as two different trigonometric points, one with a positive value for θ and one with a negative value for θ. Illustrate the following on a unit circle diagram. a cos ( ) b sin (5 ) c cos ( ) d sin ( 9 ) Illustrate the following on a unit circle diagram. 5 a sin b cos c cos (5) d sin 5 Illustrate the following on a unit circle diagram. 5 a tan (5 ) b tan c tan d tan ( ) a Given f(t) = sin (t), use a calculator to evaluate f() to decimal places. b Given g(u) = cos (u), use a calculator to evaluate g() to decimal places. c Given h(θ) = tan (θ), use a calculator to evaluate h() to decimal places. a i On a unit circle diagram show the trigonometric point P[] and the line segments sin (), cos () and tan (). Label them with their length measures epressed to decimal places. ii State the Cartesian coordinates of P to decimal places. b On a unit circle diagram show the trigonometric points A[] and P[θ] where θ is acute, and show the line segments sin (θ) and tan (θ). B comparing the lengths of the line segments with the length of the arc AP, eplain wh sin (θ) < θ < tan (θ) for acute θ. 5 a The trigonometric point P[θ] has Cartesian coordinates (.8,.). State the quadrant in which P lies and the values of sin (θ), cos (θ) and tan (θ). b The trigonometric point Q[θ] has Cartesian coordinates,. State the quadrant in which Q lies and the values of sin (θ), cos (θ) and tan (θ). Topic 9 Trigonometric functions 8

27 Master 9.5 Units & AOS Topic Concept Smmetr properties Concept summar Practice questions Interactivit All Sin Cos Tan int-58 c For the trigonometric point R[θ] with Cartesian coordinates 5, 5, state the quadrant in which R lies and the values of sin (θ), cos (θ) and tan (θ). d The Cartesian coordinates of the trigonometric point S[θ] are (, ). Describe the position of S and state the values of sin (θ), cos (θ) and tan (θ). B locating the appropriate trigonometric point and its corresponding Cartesian coordinates, obtain the eact values of the following. a cos () b sin c tan () d cos e sin () f cos 7 + tan ( ) + sin 7 Use CAS technolog to calculate the eact value of the following. a cos 7 + sin b cos + sin c sin cos d sin (7 ) + cos (7 ) e sin (t) + cos (t); eplain the result with reference to the unit circle 8 a Obtain the eact Cartesian coordinates of the trigonometric points P 7 and Q, and describe the relative position of the points P and Q on the unit circle. b Obtain the eact Cartesian coordinates of the trigonometric points R and 5 S, and describe the relative position of these points on the unit circle. 5 c Give the eact sine, cosine and tangent values of: i 7 and, and compare the values ii 5 and, and compare the values. 5 Smmetr properties There are relationships between the coordinates and associated trigonometric values of trigonometric points placed in smmetric positions in each of the four quadrants. There will now be investigated. The signs of the sine, cosine and tangent values in the four quadrants The definitions cos (θ) =, sin (θ) =, tan (θ) = where (, ) are the Cartesian coordinates of the trigonometric point [θ] have been established. If θ lies in the first quadrant, All of the trigonometric values will be positive, since >, >. If θ lies in the second quadrant, onl the Sine value will be positive, since <, >. If θ lies in the third quadrant, onl the Tangent value will be positive, since <, <. If θ lies in the fourth quadrant, onl the Cosine value will be positive, since >, <. This is illustrated in the diagram shown. S T A C 8 Maths Quest MATHEMATICAL METHODS VCE Units and

28 WorKED EXamPLE think a Refer to the CAST diagram. There are several mnemonics for remembering the allocation of signs in this diagram: we shall use CAST and refer to the diagram as the CAST diagram. The sine, cosine and tangent values at the boundaries of the quadrants The points which do not lie within a quadrant are the four coordinate aes intercepts of the unit circle. These are called the boundar points. Since the Cartesian coordinates and the trigonometric positions of these points are known, the boundar values can be summarised b the following table. Boundar point (, ) (, ) (, ) (, ) (, ) θ radians θ degrees sin (θ) cos (θ) tan (θ) undefined undefined Other values of θ could be used for the boundar points, including negative values. a Identif the quadrant(s) where both cos (θ) and sin (θ) are negative. b If f(θ) = cos (θ), evaluate f( ). b Substitute the given value in the function rule. Identif the Cartesian coordinates of the trigonometric point. Eact trigonometric values of, and As the eact trigonometric ratios are known for angles of, 5 and, these give the trigonometric ratios for, and respectivel. A summar of these is given with the angles in each triangle epressed in radian measure. The values should be memorised. WritE a cos (θ) =, sin (θ) = The quadrant where both and are negative is quadrant. b f(θ) = cos (θ) f( ) = cos ( ) A clockwise rotation of from (, ) shows that the trigonometric point [ ] is the boundar point with coordinates (, ). Evaluate the required value of the function. The -coordinate of the boundar point gives the cosine value. cos ( ) = f( ) = Topic 9 TrigonomETric functions 8

29 θ sin (θ) or or 5 or = cos (θ) tan (θ) = = These values can be used to calculate the eact trigonometric values for other angles which lie in positions smmetric to these first-quadrant angles. Trigonometric points smmetric to [θ] where θ, 5,,,, The smmetrical points to [5 ] are shown in the diagram. Each radius of the circle drawn to each of the points makes an acute angle of 5 with either the positive or the negative -ais. The smmetric points [5 ] [5 ] to [5 ] are the endpoints of a rotation 5 5 which is 5 short of, or 5 beond, 5 5 the horizontal -ais. The calculations 8 5, and 5 give the smmetric trigonometric points [5 ], [5 ] and [5 ] respectivel. Comparisons between the coordinates of these trigonometric points with those [5 ] [5 ] of the first quadrant point [5 ] enable the trigonometric values of these non-acute angles to be calculated from those of the acute angle 5. Consider the -coordinate of each point. As the -coordinates of the trigonometric points [5 ] and [5 ] are the same, sin (5 ) = sin (5 ). Similarl, the -coordinates of the trigonometric points [5 ] and [5 ] are the same, but both are the negative of the -coordinate of [5 ]. Hence, sin (5 ) = sin (5 ) = sin (5 ). This gives the following eact sine values: sin (5 ) = ; sin (5 ) = ; sin (5 ) = ; sin (5 ) = Now consider the -coordinate of each point. As the -coordinates of the trigonometric points [5 ] and [5 ] are the same, cos (5 ) = cos (5 ). Similarl, the -coordinates of the trigonometric points [5 ] and [5 ] are the same but both are the negative of the -coordinate of [5 ]. Hence, cos (5 ) = cos (5 ) = cos (5 ). This gives the following eact cosine values: cos (5 ) = ; cos (5 ) = ; cos (5 ) = ; cos (5 ) = 8 Maths Quest MATHEMATICAL METHODS VCE Units and

30 WorKED EXamPLE think Either b considering the intercepts cut off on the vertical tangent drawn at (, ) or b using tan (θ) = sin (θ) =, ou will find that the corresponding relationships for cos (θ) the four points are tan (5 ) = tan (5 ) and tan (5 ) = tan (5 ) = tan (5 ). Hence the eact tangent values are: tan (5 ) = ; tan (5 ) = ; tan (5 ) = ; tan (5 ) = The relationships between the Cartesian coordinates of [5 ] and each of [5 ], [5 ] and [5 ] enable the trigonometric values of 5, 5 and 5 to be calculated from those of 5. If, instead of degree measure, the radian measure of is used, the smmetric points to are the endpoints of rotations which lie short of, or beond, the horizontal -ais. The positions of the smmetric points are calculated 5ο 7ο ] ] as, +,, giving the smmetric trigonometric points, 5, 7 respectivel. B comparing the Cartesian coordinates of the smmetric points with those of the first quadrant point, it is possible to obtain results such as the following selection: Second quadrant Third quadrant Fourth quadrant 5 7 cos = cos tan = tan sin = sin = = = A similar approach is used to generate smmetric points to the first quadrant points and. Calculate the eact values of the following. 5 7 a cos b sin WritE ] ] ο ] ο ο c tan ( ) ο ο ] ] ο ] a State the quadrant in which the trigonometric point lies. a cos 5 As 5 = 5 =, the point 5 lies in quadrant. Topic 9 TrigonomETric functions 85

31 Identif the first-quadrant smmetric point. S T A ] ] C ] [] Compare the coordinates of the smmetric points and obtain the required value. Note: Check the +/ sign follows the CAST diagram rule. b State the quadrant in which the trigonometric point lies. Identif the first-quadrant smmetric point. 5 ] Since 5 =, the rotation of 5 stops short of the -ais b. The points and 5 are smmetric. The -coordinates of the smmetric points are equal. 5 cos = +cos = Check: cosine is positive in quadrant. 7 b sin 7 = The point lies in quadrant. [] [ ] 7 S T A C [ ] As 7 = +, the rotation of 7 goes beond the -ais b. The points and 7 smmetric. are 8 Maths Quest MATHEMATICAL METHODS VCE Units and

32 Compare the coordinates of the smmetric points and obtain the required value. 7 is the negative of that in the first quadrant. The -coordinate of of sin 7 = sin c State the quadrant in which the trigonometric point lies. Identif the first-quadrant smmetric point. Compare the coordinates of the smmetric points and obtain the required value. Note: Alternativel, consider the intercepts that would be cut off on the vertical tangent at (, ). Interactivit Smmetr points & quadrants int-58 Smmetr properties The smmetr properties give the relationships between the trigonometric values in quadrants,, and that of the first quadrant value, called the base, with which the are smmetric. The smmetr properties are simpl a generalisation of what was covered for the bases,,. For an real number θ where < θ <, the trigonometric point [θ] lies in the first quadrant. The other quadrant values can be epressed in terms of the base θ, since the smmetric values will either be θ short of, or θ beond, the horizontal -ais. The smmetric points to [θ] are: second quadrant [ θ] third quadrant [ + θ] fourth quadrant [ θ] = Check: sine is negative in quadrant. c tan ( ) [ ] lies in quadrant. S T A C [ ] [ ] is a clockwise rotation of from the horizontal so the smmetric point in the first quadrant is [ ]. The points [ ] and [ ] have the same -coordinates but opposite -coordinates. The tangent value is negative in quadrant. tan ( ) = tan ( ) = [ θ] [ + θ] θ θ θ θ [θ] [ θ] Topic 9 Trigonometric functions 87

33 Comparing the Cartesian coordinates with those of the first-quadrant base leads to the following general statements. The smmetr properties for the second quadrant are: sin ( θ) = sin (θ) cos ( θ) = cos (θ) tan ( θ) = tan (θ) The smmetr properties for the third quadrant are: sin ( + θ) = sin (θ) cos ( + θ) = cos (θ) tan ( + θ) = tan (θ) The smmetr properties for the fourth quadrant are: sin ( θ) = sin (θ) cos ( θ) = cos (θ) tan ( θ) = tan (θ) Other forms for the smmetric points The rotation assigned to a point is not unique. With clockwise rotations or repeated revolutions, other values are alwas possible. However, the smmetr properties appl no matter how the points are described. The trigonometric point [ + θ] would lie in the first quadrant where all ratios are positive. Hence: sin ( + θ) = sin (θ) cos ( + θ) = cos (θ) tan ( + θ) = tan (θ) The trigonometric point [ θ] would lie in the fourth quadrant where onl cosine is positive. Hence: sin ( θ) = sin (θ) cos ( θ) = cos (θ) tan ( θ) = tan (θ) For negative rotations, the points smmetric to [θ] could be given as: fourth quadrant [ θ] third quadrant [ + θ] second quadrant [ θ] first quadrant [ + θ] Using smmetr properties to calculate values of trigonometric functions Trigonometric values are either the same as, or the negative of, the associated trigonometric values of the first-quadrant base; the sign is determined b the CAST diagram. The base involved is identified b noting the rotation needed to reach the -ais and determining how far short of or how far beond this the smmetric point is. It is important to emphasise that for the points to be smmetric this is alwas measured from the horizontal and not the vertical ais. 88 Maths Quest MATHEMATICAL METHODS VCE Units and

34 WorKED EXamPLE To calculate a value of a trigonometric function, follow these steps. Locate the quadrant in which the trigonometric point lies Identif the first-quadrant base with which the trigonometric point is smmetric Compare the coordinates of the trigonometric point with the coordinates of the base point or use the CAST diagram rule to form the sign in the first instance Evaluate the required value eactl if there is a known eact value involving the base. With practice, the smmetr properties allow us to recognise, for eample, that 8 sin = sin 7 7 because 8 7 = + and sine is negative in the third quadrant. 7 Recognition of the smmetr properties is ver important and we should aim to be able to appl these quickl. For eample, to evaluate cos think: Second quadrant; cosine is negative; base is, and write: cos = cos = a Identif the smmetric points to [ ]. At which of these points is the tangent value the same as tan ( )? b Epress sin in terms of a first-quadrant value. 5 c If cos (θ) =., give the values of cos ( θ) and cos ( θ). d Calculate the eact value of the following. i tan 7 think a Calculate the smmetric points to the given point. ii sin WritE a Smmetric points to [ ] will be ± from the -ais. The points are: second quadrant [8 ] = [ ] third quadrant [8 + ] = [ ] fourth quadrant [ ] = [ ] Identif the quadrant. The point [ ] is in the first quadrant so tan ( ) is positive. As tangent is also positive in the third quadrant, tan ( ) = tan ( ). State the required point. The tangent value at the trigonometric point [ ] has the same value as tan ( ). Topic 9 TrigonomETric functions 89

35 b c d Epress the trigonometric value in the appropriate quadrant form. Appl the smmetr propert for that quadrant. Use the smmetr propert for the appropriate quadrant. b 5 sin sin is in the third quadrant. 5 5 = sin + 5 = sin 5 c ( θ) is second quadrant form. cos ( θ) = cos (θ) State the answer. Since cos (θ) =., cos ( θ) =.. Use the smmetr propert for the appropriate quadrant. θ is fourth quadrant form. cos ( θ) = cos (θ) State the answer. Since cos (θ) =., cos ( θ) =.. i Epress the trigonometric value in an appropriate quadrant form. Appl the smmetr propert and evaluate. ii Epress the trigonometric value in an appropriate quadrant form. Appl the smmetr propert and evaluate. d i tan ii 7 = tan + = tan = 7 tan = is in quadrant. sin = sin = sin = sin = sin = Eercise 9.5 PRactise Smmetr properties WE a Identif the quadrant(s) where cos (θ) is negative and tan (θ) is positive. b If f(θ) = tan (θ), evaluate f(). If f(t) = sin (t), evaluate f(.5). 9 Maths Quest MATHEMATICAL METHODS VCE Units and

36 WE Calculate the eact values of the following. a sin b tan Calculate the eact values of sin 5, cos 5 5 c cos ( ) and tan 5. Consolidate Appl the most appropriate mathematical processes and tools 5 WE a Identif the smmetric points to [75 ]. At which of these points is the cosine value the same as cos (75 )? b Epress tan 7 in terms of a first quadrant value. c If sin (θ) =.8, give the values of sin ( θ) and sin ( θ). d Calculate the eact value of the following. 5 5 i cos ii sin Given cos (θ) = p, epress the following in terms of p. a cos ( θ) b cos (5 + θ) 7 Show the boundar points on a diagram and then state the value of the following. a cos () b tan (9) c sin (7) d sin e cos 9 f tan ( ) 8 Identif the quadrant(s), or boundaries, for which the following appl. a cos (θ) >, sin (θ) < b tan (θ) >, cos (θ) > c sin (θ) >, cos (θ) < d cos (θ) = e cos (θ) =, sin (θ) > f sin (θ) =, cos (θ) < 9 Determine positions for the points in quadrants, and which are smmetric to the trigonometric point [θ] for which the value of θ is: a b c d 5 e 8 f Calculate the eact values of the following. a cos ( ) b tan (5 ) c sin ( ) d tan ( ) e cos ( 5 ) f sin (5 ) Calculate the eact values of the following. a sin d cos b tan e tan 7 Calculate the eact values of the following. c cos f sin 5 a cos b sin c tan 5 d sin 8 e cos 9 f tan Topic 9 Trigonometric functions 9

37 Master Calculate the eact values of the following. 7 a cos + cos b sin c tan 5 e cos 5 tan 5 d sin f tan sin + sin + sin 9 cos ( 7) Given cos (θ) =.9, sin (t) =. and tan () =.7, use the smmetr properties to obtain the values of the following. a cos ( + θ) b sin ( t) c tan ( ) d cos ( θ) e sin ( t) f tan ( + ) 5 If sin (θ) = p, epress the following in terms of p. a sin ( θ) b sin ( θ) c sin ( + θ) d sin (θ + ) a Verif that sin 5 + cos 5 =. b Eplain, with the aid of a unit circle diagram, wh cos ( θ) = cos (θ) is true for θ = 5. c The point [ϕ] lies in the second quadrant and has Cartesian coordinates (.5,.87). Show this on a diagram and give the values of sin ( + ϕ), cos ( + ϕ) and tan ( + ϕ). d Simplif sin ( + t) + sin ( t) + sin (t + ). e Use the unit circle to give two values of an angle A for which sin (A) = sin ( ). f With the aid of the unit circle, give three values of B for which sin (B) = sin. 7 a Identif the quadrant in which the point P[.] lies. b Calculate the Cartesian coordinates of point P[.] to decimal places. c Identif the trigonometric positions, to decimal places, of the points in the other three quadrants which are smmetric to the point P[.]. 8 Consider the point Q[θ], tan (θ) = 5. a In which two quadrants could Q lie? b Determine, to decimal places, the value of θ for each of the two points. c Calculate the eact sine and cosine values for each θ and state the eact Cartesian coordinates of each of the two points. 5 9 Maths Quest MATHEMATICAL METHODS VCE Units and

38 9. Graphs of the sine and cosine functions As the two functions sine and cosine are closel related, we shall initiall consider their graphs together. Units & AOS Topic Concept 5 Sine and cosine graphs Concept summar Practice questions Interactivit Graph plotter: Sine and cosine int-97 Interactivit The unit circle: Sine and cosine graphs int-55 The graphs of = sin () and = cos () The functions sine and cosine are both periodic and have man-to-one correspondences, which means values repeat after ever revolution around the unit circle. This means both functions have a period of since sin ( + ) = sin () and cos ( + ) = cos (). The graphs of = sin () and = cos () can be plotted using the boundar values from continued rotations, clockwise and anticlockwise, around the unit circle. sin () cos () The diagram shows four ccles of the graphs drawn on the domain [, ]. The graphs continue to repeat their wavelike pattern over their maimal domain R; the interval, or period, of each repetition is. = cos () = sin () The first observation that strikes us about these graphs is how remarkabl similar the are: a horizontal translation of to the right will transform the graph of = cos () into the graph of = sin (), while a horizontal translation of to the left transforms the graph of = sin () into the graph of = cos (). Recalling our knowledge of transformations of graphs, this observation can be epressed as: cos = sin () sin + = cos () The two functions are said to be out of phase b or to have a phase difference or phase shift of. Both graphs oscillate up and down one unit from the -ais. The -ais is the equilibrium or mean position and the distance the graphs oscillate up and down from this mean position to a maimum or minimum point is called the amplitude. The graphs keep repeating this ccle of oscillations up and down from the equilibrium position, with the amplitude measuring half the vertical distance between maimum and minimum points and the period measuring the horizontal distance between successive maimum points or between successive minimum points. Topic 9 Trigonometric functions 9

39 One ccle of the graph of = sin () The basic graph of = sin () has the domain [, ], which restricts the graph to one ccle. The graph of the function f : [, ] R, f() = sin () is shown. Ke features of the graph of = sin (): Equilibrium position is the -ais, the line with equation =. Amplitude is unit. Period is units. Domain is [, ]. Range is [, ]. The -intercepts occur at =,,. Tpe of correspondence is man-to-one. The graph lies above the -ais for (, ) and below for (, ), which matches the quadrant signs of sine given in the CAST diagram. The smmetr properties of sine are displaed in its graph as sin ( θ) = sin (θ) and sin ( + θ) = sin ( θ) = sin (θ). a a θ ), ), ) θ = sin () = sin () + θ ) θ 9 Maths Quest MATHEMATICAL METHODS VCE Units and

40 One ccle of the graph of = cos () The basic graph of = cos () has the domain [, ], which restricts the graph to one ccle. The graph of the function f : [, ] R, f() = cos () is shown. = cos () (, ) (, ) Ke features of the graph of = cos (): (, ) Equilibrium position is the -ais, the line with equation =. Amplitude is unit. Period is units. Domain is [, ]. Range is [, ]. The -intercepts occur at =,. Tpe of correspondence is man-to-one. The graph of = cos () has the same amplitude, period, equilibrium (or mean) position, domain, range and tpe of correspondence as the graph of = sin (). Guide to sketching the graphs on etended domains There is a pattern of 5 points to the shape of the basic sine and cosine graphs created b the division of the period into four equal intervals. For = sin (): first point starts at the equilibrium; the second point at of the period, reaches up one amplitude to the maimum point; the third point, at of the period, is back at equilibrium; the fourth point at of the period goes down one amplitude to the minimum point; the fifth point at the end of the period interval returns back to equilibrium. In other words: equilibrium range maimum equilibrium range minimum equilibrium For = cos (): the pattern for one ccle is summarised as: range maimum equilibrium range minimum equilibrium range maimum This pattern onl needs to be continued in order to sketch the graph of = sin () or = cos () on a domain other than [, ]. Topic 9 Trigonometric functions 95

41 WorKED EXamPLE Sketch the graph of = sin () over the domain [, ] and state the number of ccles of the sine function drawn. think Draw the graph of the function over [, ]. Etend the pattern to cover the domain specified. State the number of ccles of the function that are shown in the graph. WritE The basic graph of = sin () over the domain [, ] is drawn. = sin () 5 7,, ) graphical and numerical solutions to equations The sine and cosine functions are transcendental functions, meaning the cannot be epressed as algebraic epressions in powers of. There is no algebraic method of solution for an equation such as sin () = because it contains a transcendental function and a linear polnomial function. However, whether solutions eist or not can usuall be determined b graphing both functions to see if, or in how man places, the intersect. If a solution eists, an interval in which the root lies could be specified and the bisection method could refine this interval. When sketching, care is needed with the scaling of the -ais. The polnomial function is normall graphed using an integer scale whereas the trigonometric function normall uses multiples of. It can be helpful to remember that., so.57 and so on. ) ) ) The pattern is etended for one ccle in the negative direction and one further ccle in the positive direction to cover the domain,., 5 7 (, ),, = sin () ) ) ) 5, ) ) ) ) 7, Altogether, ccles of the sine function have been drawn. ) ) ) 9 maths QuEsT mathematical methods VcE units and

42 WorKED EXamPLE 5 Consider the equation sin () =. think a Calculate the points needed to sketch the two graphs. a Sketch the graphs of = sin () and = on the same set of aes and eplain wh the equation sin () = has onl one root. b Use the graph to give an interval in which the root of the equation sin () = lies. c Use the bisection method to create two narrower intervals for the root and hence give an estimate of its value. Sketch the graphs on the same set of aes. Give an eplanation about the number of roots to the equation. b State an interval in which the root of the equation lies. c Define the function whose sign is to be tested in the bisection method procedure. Test the sign at the endpoints of the interval in which the root has been placed. WritE a = sin () One ccle of this graph has a domain [, ]. The ais intercepts are (, ), (, ) and (, ). = has ais intercepts at (, ) and (.5, )..5 = sin () = The two graphs intersect at one point onl so the equation sin () = has onl one root. b From the graph it can be seen that the root lies between the origin and the -intercept of the line. The interval in which the root lies is therefore [,.5]. c sin () = sin () + = Let f() = sin () + At =, the sine graph lies below the line so f() <. Check: f() = sin () + () = < At =.5, the sine graph lies above the line so f(.5) >. Check: f(.5) = sin (.5) + (.5) = sin (.5 c ) + =.8 > Topic 9 TrigonomETric functions 97

43 Create the first of the narrower intervals. Approimations to the sine and cosine functions for values close to zero Despite the sine and cosine functions having no algebraic form, for small values of it is possible to approimate them b simple polnomial functions for a domain close to zero. Comparing the graphs of = sin () and = we can see the two graphs resemble each other for a = domain around =. For small, sin (). This offers another wa to obtain an estimate of the root of the equation sin () =, as the graphs drawn in Worked Eample 5 placed the root in the small interval [,.5]. Replacing sin () b, the equation becomes =, the solution to which is = or... To -decimal-place accurac, this value agrees with that obtained with greater effort using the bisection method. The graph of the cosine function around = suggests a quadratic polnomial could approimate cos () for a small section of its domain around =. Midpoint of [,.5] is =.5. f(.5) = sin (.5) + (.5) =.5 < The root lies in the interval [.5,.5]. Create the second interval. Midpoint of [.5,.5] is =.75. f(.75) = sin (.75) + (.75) =. > The root lies in the interval [.5,.75]. 5 State an estimated value of the root of the equation. The midpoint of [.5,.75] is an estimate of the root. An estimate is =.5. (, ) (, ) = sin () = cos () =.5 98 Maths Quest MATHEMATICAL METHODS VCE Units and

44 Comparing the graphs of = cos () and =.5 for small, show the two graphs are close together for a small part of the domain around =. For small, cos ().5 WorKED EXamPLE Returning to the unit circle definitions of the sine and cosine functions, the line segments whose lengths give the values of sine and cosine are shown in the unit circle diagram for a small value of θ, the angle of rotation. The length PN = sin (θ) and for small θ, (, ) this length is approimatel the same as the length of the arc which subtends the angle θ at the centre of the unit circle. The arc length is rθ = θ = θ. Hence, for small θ, sin (θ) θ. The length ON = cos (θ) and for small θ, ON, the length of the radius of the circle. Hence, for small θ, cos (θ). As the polnomial.5 for small, this remains consistent with the unit circle observation. a Use the linear approimation for sin () to evaluate sin ( ) and compare the accurac of the approimation with the calculator value for sin ( ). b Show there is a root to the equation cos () = for which.. c Use the quadratic approimation.5 for cos () to obtain an estimate of the root in part b, epressed to decimal places. think WritE a Epress the angle in radian mode. a = 8 = c Use the approimation for the trigonometric function to estimate the value of the trigonometric ratio. sin () for small values of. As is small, sin c sin ( ) (, ) O θ (, ) N P[θ] (, ) Compare the approimate value with that given b a calculator for the value of the trigonometric ratio. From a calculator, sin ( ) =.5 and =.5 to 5 decimal places. The two values would be the same when epressed correct to decimal places. Topic 9 TrigonomETric functions 99

45 b c Show there is a root to the equation in the given interval. Use the approimation given to obtain an estimate of the root in part b. Eercise 9. PRactise Graphs of the sine and cosine functions WE Sketch the graph of = cos () over the domain, and state the number of ccles of the cosine function drawn. On the same set of aes, sketch the graphs of = cos () and = sin () over the domain [, ] and shade the region (, ) : sin () cos (), [, ]. WE5 Consider the equation sin () =. a Sketch the graphs of = sin () and = on the same set of aes and eplain wh the equation sin () = has onl one root. b Use the graph to give an interval in which the root of the equation sin () = lies. c Use the bisection method to create two narrower intervals for the root and hence give an estimate of its value. Consider the equation cos () =. b cos () = Let f() = cos () f() = cos () () = > f(.) = cos (.) (.) =.9989 < Therefore, there is a root of the equation for which.. c Values of in the interval. are small and for small, cos ().5. Let cos () =.5 The equation cos () = becomes: =.5 =.5 = 5 = 5 = ± 5 = ±.88 The root for which is positive is.88 to decimal places. a Give the equations of the two graphs whose intersection determines the number of solutions to the equation. 5 Maths Quest MATHEMATICAL METHODS VCE Units and

46 Consolidate Appl the most appropriate mathematical processes and tools b Sketch the two graphs and hence determine the number of roots of the equation cos () =. c For what value of k will the equation cos () = + k have eactl one solution? 5 WE a Use the linear approimation for sin () to evaluate sin (.8 ) and compare the accurac of the approimation with the calculator value for sin (.8 ). b Show there is a root to the equation cos ().5 = for which.. c Use the quadratic approimation.5 for cos () to obtain an estimate of the root in part b, epressed to decimal places. a Evaluate cos (.5) using the quadratic polnomial approimation for cos () and compare the value with that given b a calculator. b Eplain wh the quadratic approimation is not applicable for calculating the value of cos (5). 7 Sketch the graphs of = sin () and = cos () over the given domain interval. a = sin (), b = cos (), c = cos (), d = sin (),. 8 a State the number of maimum turning points on the graph of the function f : [, ] R, f() = sin (). b State the number of minimum turning points of the graph of the function f : [, ] R, f() = cos (). c Sketch the graph of = cos (), 5 and state the number of ccles of the cosine function drawn. 9 State the number of intersections that the graphs of the following make with the -ais. a = cos (), 7 b = sin (), c = sin (), d = cos (), a The graph of the function f : [, a] R, f() = cos () has intersections with the -ais. What is the smallest value possible for a? b The graph of the function f : [b, 5] R, f() = sin () has turning points. If f(b) =, what is the value of b? c If the graph of the function f : [ c, c] R, f() = sin () covers.5 periods of the sine function, what must the value of c be? a Draw one ccle of the cosine graph over [, ] and give the values of in this interval for which cos () <. b Eplain how the graph in part a illustrates what the CAST diagram sas about the sign of the cosine function. a Determine the number of solutions to each of the following equations b drawing the graphs of an appropriate pair of functions. i cos () = ii cos () = iii sin () + = Topic 9 Trigonometric functions 5

47 Master b For the equation in part a which has one solution, state an interval between two integers in which the solution lies and appl three iterations of the bisection method to obtain an estimate of the solution. c i For what values of a will the equation sin () + a = have no solution? ii How man solutions does the equation sin () + a = have if a =? Consider the equation sin () =. a Show the equation has solutions and state the eact value of one of these solutions. b One of the solutions lies in the interval [, ]. Use the bisection method to obtain this solution with an accurac that is correct to decimal place. c What is the value of the third solution? Use the linear approimation for sin () to calculate the following. a sin ( ) b sin c sin ( ) 9 d sin ; comment on the reason for the discrepanc with its eact value. e A person undergoing a particular tpe of ee test is looking at a round circle of radius cm on a screen. The person s ee is at a distance of cm from the centre of the circle. The two lines of sight of the person act tangentiall to the circle and enclose an angle of θ degrees. Use an approimation to calculate the value of θ in terms of and hence state the magnitude of the angle between the lines of sight, to decimal places. 5 a Use the quadratic approimation cos () to epress the value of cos as a rational number. b The solution to the equation cos () + 5 = is small. Use the quadratic approimation cos () to obtain the solution as an irrational number. Consider the equation sin () =. a Show the equation has a solution for which. and use the linear polnomial approimation for sin () to estimate this solution. b Specif the function whose intersection with = sin () determines the number of solutions to the equation and hence eplain wh there would be other positive solutions to the equation. c Wh were these other solutions not obtained using the linear approimation for sin ()? d Analse the behaviour of the function specified in part b to determine how man solutions of sin () = lie in the interval [, ]. 7 a i Use a CAS technolog to obtain the two solutions to the equation sin () =. ii For the solution closer to zero, compare its value with that obtained using the linear approimation for sin (). b Investigate over what interval the approimation sin () = is reasonable. 8 Sketch the graph of = cos () for [, ] and state its period. 5 Maths Quest MATHEMATICAL METHODS VCE Units and

48 ONLINE ONLY 9.7 Review the Maths Quest review is available in a customisable format for ou to demonstrate our knowledge of this topic. the review contains: short-answer questions providing ou with the opportunit to demonstrate the skills ou have developed to efficientl answer questions without the use of CAS technolog Multiple-choice questions providing ou with the opportunit to practise answering questions using CAS technolog ONLINE ONLY Activities to access ebookplus activities, log on to Interactivities A comprehensive set of relevant interactivities to bring difficult mathematical concepts to life can be found in the Resources section of our ebookplus. Etended-response questions providing ou with the opportunit to practise eam-stle questions. a summar of the ke points covered in this topic is also available as a digital document. REVIEW QUESTIONS Download the Review questions document from the links found in the Resources section of our ebookplus. studon is an interactive and highl visual online tool that helps ou to clearl identif strengths and weaknesses prior to our eams. You can then confidentl target areas of greatest need, enabling ou to achieve our best results. Units & Trigonometric functions Sit topic test Topic 9 TrigonomETric functions 5