Applying trigonometric functions

Transcription

1 Appling trigonometric functions Sllabus Guide hapter 8 ontents 8. Degrees and radians 8. Trigonometric ratios and the unit circle 8. Trigonometric graphs 8. Trigonometric functions and applications hapter summar hapter review Sllabus subject matter Periodic functions and applications Definition of a radian and its relationship with degrees Definition of the trigonometric functions sin, cos and tan of an angle in degrees and in radians Graphs of = sin, = cos and = tan for an angle in degrees ( 0 0 ) and in radians ( ) Significance of the constants A, B, and D on the graphs of = A sin (B + ) + D and = A cos (B + ) + D Applications of periodic functions Quantitative concepts and skills alculation and estimation with and without instruments Basic algebraic manipulations Plotting points using artesian coordinates

2 Appling trigonometric functions Have ou ever wondered wh there are 90 in a right angle? This is actuall part of a seagesimal sstem, based on the number 0. This number sstem originated over 000 ears ago with the ancient Sumerians and was transmitted to us through the Bablonian and Egptian cultures. With lots of factors, a sstem based on 0 makes man fractions ver simple. We still use this sstem for time and angles in navigation, astronom and direction. However, there is another angle measure that is commonl used in mathematics and science the radian. This is used because it makes calculations, functions and formulas simpler, although its does seem a little strange at first. In this chapter, ou will be introduced to radian angle measure. 8. Degrees and radians Degree angle measure is based on the ancient Bablonian number sstem. It is more convenient for man mathematical and scientific purposes to measure angles using circular angle measure.! One radian ( c or rad) 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. It is not necessar to use the smbol c or rad after an angle to show it is in radians. All angles are assumed to be in radians unless the are shown to be measured in degrees b the degree sign. r r r It follows from the above definition that, for a circle of radius r, the radian measure of the central angle subtended b an arc of length l is found b calculating the multiple that l is of r. Etra Material Length of an arc! Radian measure of a central angle Given that l and r are in the same units, l = - r B rearranging this formula, the arc length can be calculated using the radius and central angle: l = r l r For a central angle of revolution (0 ): l = - r = r r = radians so = 0 and = 80 But. so or radian 57. l r 90 New QMaths B

3 ! Radian facts = 0 = 80 radian 57. The following multiples of are often used and should also be remembered: - = 70 = 90 = 0 = 5 = 0 We can use the fact that = 80 to assist with conversions between degrees and radians. onsider an angle measured in degrees ( d ) and radians ( r ). d = r so - 80 = Rearranging gives d = - 80 r or r = - 80 d! Radian degree conversion formulas Radians to degrees: d = - 80 r Degrees to radians r = - 80 d d r Eample hange the following to degrees. 9 a - b.07 rad 5 9 a Use the formula = = Alternative method 9 Use = = = 80 b Use the formula..07 rad = Alternative method 80 Use rad = rad = hapter 8 Appling trigonometric functions

4 Appling trigonometric functions Eample hange the following to radians. a 0 b 0.5 a Use the formula. 0 = Leave in eact form. = - b Use the formula. 0.5 = Answer in eact form. = 0.5 Alternativel, evaluate and round off..9 It is sometimes helpful to show angles at appropriate points on a unit circle, including the -ais and -ais on the diagram of a circle of radius. This is also known as locating the angle in standard position. On a artesian coordinate plane, an angle is said to be in standard position when the initial side of the angle is along the positive -ais and the verte of the angle is at the origin. Verte Initial side Terminal side For an angle of, the point P() on the circumference of the unit circle is sometimes called a trigonometric point corresponding to. As shown on the right, AOP is radians. Angle subtends arc AP, so AP must be units long. ) O P() A (, 0)! Positive and negative angles When drawn in standard position, angles are measured as positive in the anticlockwise direction and negative in the clockwise direction. + 9 New QMaths B

5 Eample Simplif the following and show the resulting angles on the unit circle. a + b a hange to a common denominator. + = = - (or 5 ) The angle is halfwa between and -. 5 P b hange to a common denominator. = - 5 = - (or 00 ) The angle is between - and. O O P 5 Two angles drawn in standard position and having a common terminal side are called coterminal angles. oterminal angles differ b an integer multiple of radians or 0.! oterminal angles If α and β are coterminal angles, α β = n = n0 where n =,,,,, 0,,,,, hapter 8 Appling trigonometric functions

6 Appling trigonometric functions Eample Which of the following pairs of angles are coterminal? a α = 5 and β = 585 b α = and β = a alculate the difference between the α β = = 70 angles. = ( )0 State the result. 5 and 585 are coterminal angles. b alculate the difference between the α β = angles. ( = - ) 0 = 0 - is not an integer. State the result. and are not coterminal angles. Additional Eercise 8. Eercise 8. Degrees and radians Find the value of each of the following angles drawn in standard position. a b c 0 00 d e f 0 Epress the following in radians, leaving our answers in terms of. a 0 b 5 c 90 d 5 e 0 f 00 g 0 h 7 i 88 Epress the following in radians, correct to decimal places where necessar. a 90 b 5 c 0 d 85 e 7 f 0 g 7.8 h 5. i 75.9 onvert the following from radians to degrees, correct to decimal place where necessar. a - 5 b - 7 c - d - e 5 f 5 g 7 h 5 i New QMaths B

7 5 onvert the following from radians to degrees, correct to decimal place where necessar. a b. c 0.8 d. e.0 f 5.5 g 0.07 h 5.55 i 7.89 Which of the following pairs are coterminal angles? a 5 and 5 b 0 and 0 c - and 7 d - and e 750 and 0 f and 5 7 Simplif the following and represent the resulting angles in standard location on the unit circle. 5 a + b - c + - d - e - f - g + - h i - Investigation Sectors and segments The area of a sector or segment of a circle can be epressed as a rule involving the angle subtended at the centre of the circle. The rule is simplest when the angle is epressed in radians. Work in groups of three or four people to find the rules. Write down the area of a circle. Write the area of a sector as a fraction of the circle involving the angle at the centre. Sector Write a rule for the area of the sector, using the angle at the Segment centre epressed in radians. Write the area of a segment as the difference between the areas of a sector and triangle. 5 Write the rule for the area of a segment. alculate the areas of some sectors and segments. Etra Material Sectors and segments 8. Trigonometric ratios and the unit circle The sine (sin), cosine (cos) and tangent (tan) trigonometric (trig) ratios have alread been defined using right-angled triangles as follows: sin = opposite hpotenuse cos = adjacent hpotenuse opposite tan = - adjacent Hpotenuse Adjacent Opposite hapter 8 Appling trigonometric functions

8 Appling trigonometric functions You can use a calculator to find decimal approimations of the values of trig ratios. It is possible to find eact values of the trig ratios for the angles 0, 5 and 0 b drawing special rightangled triangles known as standard triangles and using previous definitions, as shown below. Angle = 0 = 5 = 0 sin cos - - tan The definitions of sin, cos and tan as previousl described are restricted to angles up to 90. We can use the unit circle to etend the definitions to angles of an size. Suppose the point P() on the unit circle has coordinates (, ), so we can also write it as P(, ). Draw in a vertical line to make OBP as shown. Then OB =, BP = and OP =. O P() = P(, ) B A (, 0) Using OBP in the unit circle, we can see that: sin = = cos = = tan =! Defining trigonometric ratios using the unit circle If P() = P(, ) is a point on the unit circle found b rotating the point A(, 0) anticlockwise through an angle of, then we define: sin = cos = tan = = sin cos O P() = P(, ) A (, 0) These definitions are sometimes etended further to a circle of radius r as follows: sin = cos = tan = = sin r r cos 9 New QMaths B

9 The previous definition can be used to assign values to the trig ratios for 0 (0 ), 90, 80 and = 0 Angle = 90 = 80 - = 70 = 0 sin 0 0 cos 0 0 tan 0 Not defined 0 Not defined 90 = (0,) (, 0) (, 0) 80 = 0 = (0 = 0) 70 = (0, ) Eample 5 Use the unit circle to find the value of: a sin ( ) b sin ( 90 ) c tan 5 a Draw a sketch. (, 0) From the sketch it is clear that sin ( ) = sin. State the result. sin ( ) = 0 b Draw a sketch c From the sketch it is clear that sin ( 90 ) = sin 70. State the result Draw a sketch. sin ( 90 ) = (0, ) 5 (, 0) 5 = () +, so 5 is coterminal with. From the sketch it is clear that tan 5 = tan. State the result. tan 5 = 0 hapter 8 Appling trigonometric functions

10 Appling trigonometric functions Scientific and graphics calculators can both be used to find trigonometric ratios of angles epressed in degrees or radians. It is important to check that the calculator is in the correct mode for the angle measure before using it for evaluation. Scientific calculators normall have an indication on the displa. It usuall has deg for degrees or rad for radians. Graphics calculators normall show the angle measure in a SETUP or MODE menu. The values shown b calculators are actuall approimations, but these are sufficient for most purposes. However, it is still important to be able to evaluate trigonometric ratios using onl the acute values. We can use the definitions of trig ratios with the unit circle to work out the signs of the sin, cos and tan of an angle in an (0, ) quadrant of the circle: sin =, so sin will be positive in the quadrants where nd st is positive, i.e. the first and second quadrants. quadrant quadrant (, 0) (, 0) cos =, so cos will be positive in the quadrants where is positive, i.e. the first and fourth quadrants. rd th quadrant quadrant tan =, so tan will be positive when and have the same sign, i.e. in the first and third quadrants. (0, ) This information can be summarised as follows.! Signs of the trig ratios st quadrant: All ratios are positive (A) nd quadrant: Sin onl is positive (S) rd quadrant: Tan onl is positive (T) th quadrant: os onl is positive () The above can be remembered as the mnemonic AST or All Science Teachers are urious. S T A We have previousl seen how to use right-angled triangles to work out the values of trig ratios for acute angles. We have also seen how to determine the signs of the trig ratios for an angle in the unit circle. Now we will see how to use acute angles and the AST diagram to find the values of the trig ratios for an angle. Let s begin with the second quadrant. For : sin = b cos = a tan = b - a P( a, b) Now draw in the acute angle between the terminal side of and the -ais. This angle, β, is known as the reference angle. Because β is an acute angle, we can draw it in the first quadrant to determine the values of its trig ratios. In the first quadrant: b sin β = b cos β = a tan β = a So in the second quadrant: sin = sin β cos = cos β tan = tan β P( a, b) β β Q(a, b) 98 New QMaths B

11 This means that the values of sin, cos and tan for and β differ onl in their signs. The sign of the trig ratio can be determined using the AST diagram. Let s see how this works with real values. Eample alculate the value of cos 50 without the aid of a calculator. Draw a diagram showing 50 drawn in standard position. S A 0 50 Mark in the reference angle (0 ). Using AST, we know that cos 50 is negative. Use AST and the reference angle. cos 50 = cos 0 Use the standard triangle. = T A similar approach can be used for the third quadrant. For : b sin = b cos = a tan = a For the reference angle, β: b sin β = b cos β = a tan β = a β β Q(a, b) So in the third quadrant: sin = sin β cos = cos β tan = tan β P( a, b) Eample 7 Find the value of tan 5 without using a calculator. Draw a diagram showing 5 drawn in standard position. S A 5 5 T Mark in the reference angle (5 ). Using AST, we know that tan 5 is positive. Use AST and the reference angle. tan 5 = tan 5 Use the standard triangle. = hapter 8 Appling trigonometric functions

12 Appling trigonometric functions In the fourth quadrant, for. sin = b cos = a tan = For the reference angle, β: sin β = b cos β = a tan β = b - a b a β β Q(a, b) P(a, b) So in the fourth quadrant: sin = sin β cos = cos β tan = tan β Eample 8 Find the value of sin 0 without using a calculator. Draw a diagram showing 0 drawn in standard position Mark in the reference angle (0 ). Using AST, we know that sin 0 is negative. Use AST and the reference angle. sin 0 = sin 0 Use the standard triangle. = The procedures on the previous pages can be used to find the value of an trigonometric ratio in terms of an acute angle and are summarised below.! Trigonometric ratios for an angle of an magnitude For an angle drawn in standard position on the artesian plane, the acute angle between the terminal side of the angle and the -ais is known as the reference angle. The reference angle is alwas taken to be positive. In the diagram shown here, β is the reference angle for, which β is in the third quadrant of the artesian plane. For an angle, use the following steps to find the value of sin, cos or tan. Draw in standard position on the artesian plane. alculate the reference angle, β, that corresponds to. Use AST to determine the sign (+ or ) of the value of the trig ratio. Find the value of the required trig ratio for β and include the sign. 00 New QMaths B

13 Eample 9 Find the eact value of: a sin 5 b cos c tan - a Draw a sketch. S A 5 5 alculate the reference angle. Using AST, sin 5 is negative. Write the trig ratio using the reference angle. sin 5 = sin 5 Use the standard triangle. = b Draw a sketch. T S A c alculate the reference angle. Using AST, cos is positive. Write the trig ratio using the reference angle. cos = cos Use the standard triangle. = Draw a sketch. T S A alculate the reference angle. Using AST, tan - is negative. Use the reference angle to write the trig ratio. tan - = tan Use the standard triangle. = T hapter 8 Appling trigonometric functions

14 Appling trigonometric functions Eample 0 Find values of the following without using a calculator. a cos 570 b 5 sin c tan a Find (0 0 ) coterminal with 570. cos 570 = cos (570 0 ) = cos 0 Draw a sketch. S A 0 0 alculate the reference angle. Using AST, cos 570 (or cos 0 ) is negative. Write the trig ratio. cos 0 = cos 0 Use cos 570 = cos 0. cos 570 = 5 5 b Find (0 ) coterminal with. sin = sin 5 = sin Draw a sketch. T + S A alculate the reference angle. 5 Using AST, sin or sin is negative. Write the trig ratio. sin = sin 5 Use sin = sin 5. sin = c Find (0 ) coterminal with. tan = tan 8 - = tan - Use the table of values. tan - is not defined. Use tan = tan -. The value of tan is not defined. T 0 New QMaths B

15 The same procedure can used to simplif epressions involving trig ratios. Eample Simplif sin + sin ( ) where 0. Draw a sketch for. S T A The reference angle is. ( ) is in the th quadrant, so sin ( ) is negative. Use the reference angle. sin ( ) = sin Substitute into the given epression. sin + sin ( ) = sin sin Simplif. = sin The right-angled triangle formed b the reference angle in the unit circle is known as the reference triangle.! Reference triangle The terminal side of the reference angle β meets the unit circle at P(a, b), where a or b ma be negative. The reference triangle is formed b drawing a perpendicular from P to the -ais. In the diagram at right, POM is the reference triangle for β. Using Pthagoras s theorem we can see that a + b = M a P(a, b) b β O This definition can be etended beond the unit circle reference angles. For the reference triangle POM, a + b = r So r = a + b O a β r M b P(a, b) hapter 8 Appling trigonometric functions

16 Appling trigonometric functions Eample The reference angle φ passes through P(5, ). Use the reference triangle to find an eact value for sin φ. Draw a diagram and mark in the reference triangle. O 5 φ M Use Pthagoras s theorem. OP = 5 + ( ) OP is positive. = Use the definition of sin. sin φ = - P (5, ) Additional Eercise 8. Eercise 8. Trigonometric ratios and the unit circle Use the unit circle to find: a cos 0 b sin 0 c tan 5 d cos 00 e tan 0 f cos 0 Use the unit circle to find: 7 7 a cos - b sin - c cos d tan e sin - f cos - Use the unit circle to find: a sin 90 b cos c tan d cos - e tan 80 f tan 50 Use the unit circle to find: a sin 0 b cos - 7 c tan - d cos e tan 0 f cos 5 5 If is drawn in standard position on the artesian plane, in which quadrants must the terminal side of lie for the following to be true? a cos 0 b tan 0 c sin 0 d tan 0 e sin 0 f cos 0 a What is the greatest value of in the unit circle? b What is the smallest value of in the unit circle? c What is the range of sin? 7 a What is the greatest value of in the unit circle? b What is the smallest value of in the unit circle? c What is the range of cos? 0 New QMaths B

17 8 a Does have a greatest or least value when and are in the unit circle? b What is the range of tan? c Are there an values of for which is not defined? d What is the domain of tan? 9 In each of the following cases, verif that P lies on the unit circle. If is an angle drawn in standard position and its terminal side passes through P, find the eact values of sin, cos and tan without actuall calculating the value of. a P, b P c P , -, d P, e P f P,, 0 In each of the following cases, is an angle drawn in standard position. Without actuall calculating the value of, find the eact values of the other two trig ratios using the information supplied. a sin = and is in quadrant b cos = - 5 and is in quadrant 5 c 7 cos = - and is in quadrant d sin = and is in quadrant 5 5 e sin = and is in quadrant f cos = and is in quadrant g cos = - and is in quadrant h sin = - and is in quadrant 5 In each of the following cases, the reference angle β passes through point P. Use the reference triangle to find the eact values of sin β and cos β. a P(, ) b P( 5, ) c P(7, ) d P(, ) e P(, ) f P(, 5) Find the eact values of: a sin 5 b cos 50 c tan 0 d cos 5 e tan 5 f sin 0 g tan 0 h cos 00 i sin 0 Find the eact values of: a 7 sin - 5 b cos - c tan - d cos - e tan 7 f sin - 5 g sin h cos - i tan - j cos k sin 5 l sin - Find the eact values of: a sin b 7 cos - c sin - d cos - e 7 cos - f tan g sin - h cos - i 5 sin - 7 j tan - 7 k tan - l sin hapter 8 Appling trigonometric functions

18 Appling trigonometric functions 5 Find the eact values of: 7 5 a sin b cos c tan d cos - 9 e sin 50 f tan g tan 7 h sin - i cos 00 j cos 570 k tan 90 l sin 90 Modelling and problem solving Simplif the following. a tan ( + ) + tan b cos ( ) + cos ( ) c sin ( + 5) 5 sin 5 d cos ( + ) + cos ( ) e sin ( + ) sin f cos ( + g) cos ( g) g tan (q + ) + tan (q ) h sin (m + ) + sin (m ) i sin ( c) + sin ( + c) 8. Trigonometric graphs The graphs of the sine, cosine and tangent functions ma be drawn b using their definitions and knowledge of the unit circle. Graph of the sine function onstruct a unit circle on some graph paper, and on its circumference mark P -, P, P,, P b drawing diameters as shown in the diagram below. onstruct artesian aes with running from to and from to aligned with the unit circle. Mark the -ais in units of - as shown. Now, sin = = -coordinate of P. So the graph of = sin can be drawn b transferring the -coordinates of the points P() on the unit circle to the artesian aes. This can be done phsicall as shown. = sin The graph above has been etended before 0 and past using the smmetr of a circle or periodic nature of the sine function. The sign of the graph (whether it is above or below the ais) also matches the previous work on the sign of the trig ratios in each quadrant, as shown at the top of the following page. 0 New QMaths B

19 +S +A T Graph of the cosine function A similar method can be used to draw a graph of the cosine function, = cos. However, the lengths of the -coordinates must be used for cos, so it is necessar to transfer the lengths to the graph using a pair of dividers. = cos The resulting graph of = cos shares man similar features with the graph of = sin. Graph of the tangent function To draw the graph of the tangent function, we begin as for the sine and cosine ecept that the tangent, AT, is drawn to touch the unit circle at A. OP is etended to meet the tangent at T. Now OMP OAT So = - AT = AT Thus tan = AT So tan = length of tangent cut off b the etended radius. That is, tan = -coordinate of T. O P T M A In the fourth quadrant, this is negative. In the second and third quadrants, the radius is etended backwards to meet the tangent. T A A A T T hapter 8 Appling trigonometric functions

20 Appling trigonometric functions For convenience, it is best to start plotting the graph = tan between and. = tan B looking ate the graphs of = sin, = cos and = tan, ou can see that these are all periodic functions. Periodic functions were discussed in hapter. You can also see that = cos and = sin have the same period and amplitude. A graphics calculator can also be used to draw graphs of the trigonometric functions, as alread seen in hapter. Eample Use a graphics calculator to draw the graph of = sin from = to. Use SETUP or MODE to set the calculator to radian measure. Put the function in as Y = SIN ( X,T,,n ). Set the WINDOW (or V-Window) so that X goes from to in units of, and so that Y goes from to in units of 0.. GRAPH the function to obtain: asio f-980g AU Teas Instruments TI-8 Sharp EL-9900 Additional Eercise 8. Teacher Notes Eercise 8. Trigonometric graphs For questions to, ou ma wish to print and use the blank grid on the D-ROM. Use a unit circle and graph paper to draw the graph of = sin from = to. Use a unit circle and graph paper to draw the graph of = cos from = to. Use a unit circle and graph paper to draw the graph of = tan from = to. 08 New QMaths B

21 Use the following graphs of = sin and = cos to answer the questions below. = sin = cos a What are the periods and amplitudes of the sine and cosine functions? b omment on the similarities and differences of the graphs. 5 a Use a graphics calculator to draw the graph of = tan from = to. b What are the period and amplitude of the tangent function? 8. Trigonometric functions and applications To emphasise the fact that the variable is considered to be a real number instead of an angle, we use the smbol instead of in most applications of trigonometric functions. The nature of the basic graphs is summarised below and on the net page.! Basic trigonometric functions = sin Domain = real numbers Range is. Period = Amplitude = Zeros are at n:,,, 0,,,,, ( Maima are at n + ) :, -, 5, -, ( Minima are at n + ) :,, - 7, -, = cos Domain = real numbers Range is. Period = Amplitude = ( Zeros are at n + ) :, -,,, - 5, - 7, - Maima are at n:,, 0,,, Minima are at (n + ):,,,, = cos = sin hapter 8 Appling trigonometric functions

22 Appling trigonometric functions = tan Domain = real numbers ecept odd multiples of Range = real numbers Period = Amplitude = infinite = tan Zeros are at n:,,, 0,,,,, ( The graph of tan has asmptotes at n + ) :, - 5 7,,, -, -, -, At these values, the function is undefined, but near them the graph becomes ver large or ver small. Some trigonometric function graphs have been shown in hapter as eamples of periodic functions. In this section a more sstematic stud is made of the sine and cosine functions. Eample Make a table of values and hence sketch the graph of = sin from to. omment on the relationship with the graph of = sin. Set up the table with values at ever. Round values of sin to decimal place. 5 0 sin sin Sketch the graph, using knowledge of the basic shape of = sin to complete the curve. = sin = sin The graph of = sin is stretched verticall b a factor of compared with = sin. 0 New QMaths B

23 Eample 5 Use a graphics calculator to draw the graph of = cos + from to. omment on the relationship with the graph of = cos. Enter the function as Y = OS ( X,T,,n ) +. Set the WINDOW (or V-Window) so that: X min = Y min = 0 X ma = Y ma = X scl = Y scl = Then GRAPH the function. The displa on our graphics calculator should look something like this. There will be minor variations depending on the brand of calculator, but the major features of the graph will be the same. The graph of = cos + is translated units upwards compared with the graph of = cos. Eample Make a table of values and hence sketch the graph of = cos from to. omment on its relationship with the graph of = cos. is about. and is about.. Make a table of values from. to cos cos Sketch the graph, using our knowledge of the basic shape of = cos. = cos = cos The graph of = cos is compressed horizontall b a factor of compared with the graph of = cos. The period of = cos is of the period of = cos. hapter 8 Appling trigonometric functions

24 Appling trigonometric functions Eample 7 Use a graphics calculator to compare the graphs of = sin and = sin. Enter the functions in Y and Y and set the WINDOW (or V-Window) to show X from to with an X-scale (Xscl) of - and Y from to with a Y-scale (Yscl) of 0.5. Then GRAPH the functions. = sin The displa on our graphics calculator should look something like this. There will be minor variations depending on the brand of calculator, but the major features of the graph are the same. The graph of = sin is translated - to the right compared with the graph of = sin. There is a horizontal shift of = - to the right compared with = sin. = sin ( ) After working through the previous eamples, ou should be able to see that the following rules appl to general graphs of the sine and cosine functions.! The graphs of the functions = A sin (B + ) + D and = A cos (B + ) + D are related to graphs of the functions = sin and = cos b: Vertical translation of D For D 0 it is upwards and for D 0 it is downwards. 0 = sin + = sin Horizontal translation of B For - 0 it is to the left and for - 0 it is to the right. B B = sin 0 = cos = cos = cos 0 0 = sin ( ) = cos ( + ) New QMaths B

25 Horizontal change of scale of B For B it compresses b a factor of B and for 0 B it stretches b a factor of. If B is negative, the graph is reflected in the -ais, so the values swap. B = sin = cos 0 0 = sin Vertical change of scale of A For A it stretches b a factor of A and for 0 A it compresses b a factor of -. If A is negative, the graph is inverted (reflected in the -ais). A 0 = sin = sin = cos = cos = cos! Summar of the features of sine and cosine functions For the functions = A sin (B + ) + D = A sin B D B and = A cos (B + ) + D = A cos B D B the amplitude is the magnitude of A the period is - B the value - is called the phase shift and is the horizontal translation B the average (mean) value is D.! Sketching trigonometric graphs Graphs of sine and cosine functions ma be sketched from the equation b two methods: Method : Identif the translations and changes of scale from the equation. Method : Identif the: starting point zeros end of a ccle maima and minima. Find the values of that produce the sine or cosine of 0,,, - and and the corresponding values of. hapter 8 Appling trigonometric functions

26 Appling trigonometric functions Eample 8 Without using a calculator or a table of values, sketch the graph of = sin +. Method onsider first the graph It has the basic shape of the sine function, but is of the simplified function compressed b a horizontal factor of, so the period is = sin. instead of. It is stretched verticall b a factor of, so the amplitude is. The graph of = sin is shown in green below. Now consider the translations It is translated to the left b a phase shift of - to obtain (compared with = sin ). = sin + It is translated downwards, so the average value is. The final graph is shown in blue below. = sin -. Method Find (, ) when we get sin 0 = 0. + = 0 = - = sin 0 = The first ccle starts at -,. Find (, ) so we get sin =. + = =, so the point is,. Find (, ) so we get sin = 0. + = 5 = -, so the point is - 5,. Find (, ) so we get sin - =. + = - = -, so the point is -, 7. Find (, ) so we get sin = 0. + = =, so the point is,. The five points are shown on the first ccle graph below in blue. = sin = sin ( + ) - New QMaths B

27 Eample 9 Without using a calculator or a table of values, sketch the graph of = cos -. heck our sketch with the aid of a graphics calculator. Method onsider first the graph It has the basic shape of the cosine function, but is of the simplified function compressed b a horizontal factor of, so the period = cos. is - instead of. It is stretched verticall b a factor of, so the amplitude is. The graph is inverted from the cosine graph because of the negative sign. The graph of = cos is shown in green below. onsider the translations to obtain It is translated to the right b a phase shift of - (compared with = cos ). = cos - It is translated upwards, so the average value is. = cos -. The final graph is shown in blue below. Method Find (, ) so we get cos 0 =. - = 0 = -, so the first point is -,. Find (, ) so we get cos = 0. = - 7 = -, so the point is - 7,. Find (, ) so we get cos =. - = =, so the point is,. Find (, ) so we get cos - = 0. = =, so the point is 9,. Find (, ) so we get cos =. - = 5 =, so the point is 5,. The five points are shown on the first ccle graph below in blue. = cos ( + - ) = cos hapter 8 Appling trigonometric functions

28 Appling trigonometric functions Your graphics calculator should show similar results for Eamples 7 and 8, allowing for differences in scales. The sine and cosine functions can be used to model a wide range of periodic phenomena to a high degree of accurac. A sinusoidal model has the shape of the sine graph. This can be sine or cosine, whichever is more convenient. Eample 0 In a harbour, high tides occur h 0 min apart. The depth of water at the entrance bar is.8 m at low tide and 8. m at high tide. A high tide occurs at am on October. a Sketch a rough version of the tidal heights. b Find a sinusoidal model for the depth of the water, d, as a function of the time t hours since midnight on October. c Find the depth of water at pm on 5 October. d Use a graphics calculator to find the times on October at which a ship with a draught of 5 m can enter the harbour. a Low tides occur halfwa between high tides, so the net low tide will be h 0 min after am, at 8:0 am. The average depth will be halfwa between the tides, so the first average depth will occur h 5 min after am, which is 5:05 am. The depth will be (8. +.8) = 5. m. We can continue adding h 5 min to the last times to get the following times and depths. Time :00 am 5:05 am 8:0 am :5 am :0 pm 5:5 pm 8:0 pm :5 pm d (m) Draw a sketch using the points found and knowledge of the shape of sine and cosine. d (m) t (h) b Since the graph starts near a maimum, it is more appropriate to use cosine than sine. Write the cosine function. = A cos B D B Find the amplitude. Amplitude = (8..8) m A =.8 m Find the period. Period = h 7 = - h New QMaths B

29 7 Use period = -. - = - B B Rearrange to find B. B = - 7 Find the phase shift. Maimum is hours after t = 0. Phase shift - B = h Find the mean value. D = (8. +.8) = 5. m hoose variables. t =, d = Use = A cos B D. d =.8 cos (t ) + 5. B - 7 c At pm on 5 October, t = h. d =.8 cos - ( ) m d Enter the function in Y. Y =.8 cos ( 7 (X )) + 5. Use VARS to enter Y. Y = Y 5 Turn Y off: For the asio, use the SEL F ke to select Y. For the Teas Instruments, place the cursor over Y and press ENTER. For the Sharp, do the same as for the Teas Instruments. GRAPH the function Y. The displa on our graphics calculator should look something like this. There will be minor variations depending on the brand of calculator, but the major features of the graph will be the same. Find the zeros of Y : For the asio, use ROOT in G-Solv, and press the right cursor to get each value. For the Teas Instruments, use :zero in AL and appropriate guesses to get each value. For the Sharp, use 5 X_Incpt in AL repeatedl to get each value. Using AL or G-Solv gives X = 5.507, X = , X = , X =.59 hange to times and write answers. The ship could enter from midnight to 5:0 am, from 0:50 am to 5:50 pm, and after :0 pm. An alternative method is to draw graphs of Y =.8 cos ( 7 (X )) + 5. and Y = 5 using a graphics calculator, then find the intersections of the graphs in a similar wa to the zeros. hapter 8 Appling trigonometric functions

30 Appling trigonometric functions Investigation Musical beats To tune a guitar, the frets are used to pla a note on one string that should be the same as the base note of the net string. When the strings are in tune, the notes will be the same. When the strings are quite out of tune, the sound quite different. When the are close, but not quite the same, the sound will seem to beat (fade and strengthen rhthmicall). This can be shown quite easil on a graphics calculator. onsider a note that should have a frequenc of 0 Hz. Store this in the memor in. Store in the memor in D. Put in Y = sin ( X), Y = sin ( D X) and Y = Y + Y, and turn Y and Y off. Set the WINDOW (or V-Window) with X from 0 to 5 and Y from to. GRAPH the result. hange D to and graph it again. hange D to 0., change X to go from 0 to 0 and graph again. Find what note corresponds to 0 Hz. Tr different values of and D. What do ou find? Write a short account of our results. Additional Eercise 8. Eercise 8. Trigonometric functions and applications For 0, sketch a graph of each of the following and check with a graphics calculator. a = sin b = sin c = cos d = sin + For, sketch a graph of each of the following and check with a graphics calculator. a = sin b = sin c = cos d = sin + Sketch a graph of each of the following and check with a graphics calculator, showing a full ccle. a = sin b = cos c = 5 sin d = sin 8 + Sketch graphs of the following and check with a graphics calculator, showing a full ccle of each. a = sin b = sin + c = cos d = cos + e = sin f = sin + g = cos h = cos + 5 Sketch graphs of the following and check with a graphics calculator, showing a full ccle of each. a = sin + b = cos + c = 5 sin + 8 d = cos + e = cos + f = sin 8 New QMaths B

31 Sketch graphs and check with a graphics calculator, showing a full ccle of each. a = sin + b = cos ( + ) c = sin + - d = 5 cos - e = 5 cos f = sin + Modelling and problem solving 7 The depth of water, d(t) m, at a pier changes according to the rule d(t) = 5 + cos t where t is time in hours from high tide. On 8 March, high tide was at 7:0 am. a Draw a sketch showing the depth of water for the hours after 7:0 am on 8 March. b When was the first low tide on that da? c When was the water at a depth of m? d How long is it between low tide and high tide? t 8 The sales, S in 00s of units, of a seasonal product are modelled b S = cos where t is the time in months (t = is Januar and t = is December). a Draw a graph of the sales for a period of months. b Use the graph to determine the months for which sales eceed 800 units. 9 In an unusual meteorological investigation, the temperature, T, in a town in central Queensland was found to fluctuate approimatel according to the rule T = 5 + sin 0.t where t is the number of hours after 0:00 am. a Sketch a graph of the temperature fluctuations for a sufficient number of hours to be able to determine the maimum and minimum temperatures for that da and the net night. b Use the graph to determine the maimum and minimum temperatures. c When did the occur? d At what time was the temperature: i 7? ii 0? e Wh was this unusual? 0 In alcutta, the highest mean monthl temperature is 9.5 in June and the lowest is 8. in December. Find a model for the temperature throughout the ear and graph it. In Upernivik, Greenland, the average temperature varies between in Januar and 5 in Jul. Find and graph a model for the average temperature. P A large hoop is rolling along the ground. The vertical distance above the ground of a point P on the rim of the hoop is given b =.5.5 cos t where is in metres and t is in seconds. Find the first three times at which P is.00 m above the ground. The time between successive high tides at a pier is h 0 min. The average depth of water is m, but at low tide it is. m. Write an equation to model this relationship and use it to find the lengths of time that a boat with a draught of.5 m and.5 m can use the pier. An oscilloscope tracing signals from a microphone shows a sinusoidal curve. The amplitude relates to the volume of the sound while the frequenc gives the pitch. A violin string vibrates at 00 Hz ( Hz is vibration per second). The oscilloscope trace amplitude is V. a Find the period of the sound wave. b Write an equation for the sound wave. hapter 8 Appling trigonometric functions

32 hapter summar The radian measure ( c or rad) of a central angle is the ratio of the l arc length (l) to the length of the radius (r) subtended the centre of the circle. l = - or l = r r r Radians can be converted to degrees using formulas: 80 Radians to degrees: d = - Degrees to radians r = r - 80 d Some radian to degree conversions should be committed to memor: = 0 - = 70 = 80 = 90 = 0 = 5 = 0 On a artesian coordinate plane, an angle is in standard position when the initial side of the angle is along the positive -ais and the verte of the angle is at the origin. When drawn in standard position, angles are measured as positive in the anticlockwise direction and negative in the clockwise direction. oterminal angles differ b an integer multiple of radians or 0. The trigonometric ratios ma be defined in terms of the P() = P(, ) unit circle as: sin = = -coordinate of P cos = = -coordinate of P A (, 0) O B tan = = - -coordinate of P -coordinate of P Standard triangles ma be used to find the trigonometric ratios for the angle 5, 0 and 0. AST can be used to determine the signs of trig ratios: st quadrant: All ratios are positive (A) nd quadrant: Sin onl is positive (S) rd quadrant: Tan onl is positive (T) th quadrant: os onl is positive () S A T The reference angle (β) is the acute angle between the terminal side of an angle drawn in standard position () and the -ais. β 0 New QMaths B

33 hapter summar The value of sin, cos or tan for an angle, ma be found b: Drawing in standard position. alculating the reference angle β: Using AST to determine the sign (+ or ). Finding the value of the trig ratio for β including the sign. POM is the reference triangle for reference angle β. For the reference triangle POM, a + b = r So r = a + b O a β r M b P (a, b) For the functions = sin and = cos : domain = real numbers range is period = amplitude = = sin = cos For the function = tan : domain = real numbers, ecept odd multiples of range = real numbers period = amplitude is infinite. = tan For the functions = A sin (B + ) + D = A sin B D B and = A cos (B + ) + D = A cos B D: B the amplitude is A the period is - B the value - is called the phase shift and is the horizontal translation B the average (mean) value is D and is the vertical translation. Sine and cosine graphs ma be sketched using translations and changes of scale or b finding the five important points in a ccle. hapter 8 Appling trigonometric functions

34 hapter review E 8. Knowledge and procedures Find the value of each of the following angles drawn in standard position. a b c 0 d e f 5 5 E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. Epress the following in radians, leaving our answers in terms of. a 70 b 5 c 0 d 0 e 0 f 88 5 On a unit circle, an arc of length - subtends an angle at the centre of: A 0 B 5 0 D 50 E 0. c in degrees is given b: 80 A. - B. -. D. E 80 5 Epress in radians correct to decimal places. a 7 b 0.75 Epress in degrees, correct to decimal place. a - b 0. 7 Which angle is larger: or 0? 8 Which of the following pairs are coterminal angles? 5 7 a 8 and 5 b 0 and 500 c - and - 9 Simplif the following and represent the resulting angles in standard location on the unit circle. 5 a + - b c If is drawn in standard position on the artesian plane, in which quadrants must the terminal side of lie for the following to be true? a cos 0 b tan 0 c sin 0 Draw unit circles and use them to find sin 70 and cos 0. Draw standard triangles and use them to find tan 7 and sin -.. New QMaths B

35 hapter review 5 Show the angle - on a unit circle and give the third- and fourth-quadrant angles with cosines of the same magnitude. The eact value of sin - is: A 0 B 0 D E 5 cos ( + ) = A cos B sin cos D sin E + cos Find the eact values of: a sin 0 b cos 0 c tan d sin - e cos - f tan - 7 In each of the following cases, verif that P lies on the unit circle. If is an angle drawn in standard position and its terminal side passes through P, find the eact values of sin, cos and tan without actuall calculating the value of. a P b P 5, 5-5, - 8 In each of the following cases, is an angle drawn in standard position. Without actuall calculating the value of, find the eact value of the other two trig ratios using the information supplied. 5 a sin = - and is in quadrant b cos = and is in quadrant 5 9 In each of the following cases, the reference angle β passes through point P. Use the reference triangle to find the eact values of sin β and cos β. a P(, ) b P(, 5) c P( 7, ) 0 alculate the amplitude of the function = cos 7. The graph with the rule = sin + has period: A B D - E alculate the range of the function = cos. The equation of the graph on the right could be: E = sin A = sin B = sin = sin D = sin 0 The amplitude of = 7 cos + is: A 7 B D 0 E 5 = sin has amplitude and period: A, B,, D, E 5, Sketch the graph of each of the following, showing a full ccle. a = sin b = cos c = sin d = cos + e = 5 sin + f = cos + E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. E 8. hapter 8 Appling trigonometric functions

36 hapter review E 8. E 8. E 8. E 8. E 8. E 8. E 8. 7 The depth of water in a harbour is modelled b t D =.5 sin +.5 where t is in hours. The number of hours between high tides is: A - B D 7 E 8 The temperature in a room varies over the da in a sinusoidal fashion, with a minimum of 8 at am and a maimum of 0 at pm. a State the epected temperature at noon. b At what other time(s) of the da would ou epect the temperature to be the same as at noon? c What is the period of the temperature variations? 9 The voltage of an A current that is nominall A volts is given b: V = A sin 00t where V is in volts and t in seconds. If A = 0: a what is the period? b what is the maimum voltage? c what is the amplitude? Modelling and problem solving 0 Simplif tan ( + ) 5 tan ( ). The monthl average overnight minimum temperature for Melbourne can be modelled b the function m T =.5 cos where T is the temperature in degrees elsius and m is the number of months since Januar, assuming the can be taken as equal in length. a What is the maimum average overnight minimum, and when does it occur? b What is the mean value of the average overnight minimum, and during what months does it occur? c During what months is the average overnight minimum increasing? The depth of water in a port is given b t d =.8 cos +. where d is in metres and t is the number of hours since high tide. a How man hours are there between high tides? b If high tide was at midnight, during what hours was the depth more than. m? c During what hours was the depth less than 0. m? A clock stands on a shelf. The centre of the clock, where the hands are attached, is 5 cm above the shelf, and the minute hand is 0 cm long. The distance of the tip of the hand from the shelf can be modelled b a sinusoidal function. Write an equation for the model in terms of the time t, where t is the number of minutes past midnight. Hence find the number of times during the da when the tip is 0 cm above the shelf. New QMaths B