MCR3U UNIT #6: TRIGONOMETRY

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Elizabeth Chapman
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1 MCR3U UNIT #6: TRIGONOMETRY SECTION PAGE NUMBERS HOMEWORK Prerequisite p. 0 - # 3 Skills 4. p. 8-9 #4, 5, 6, 7, 8, 9,, 4. p #bde, acd, 3, 4acde, 5, 6ace, 7, 8, 9, 0,, 4.3 p #aef,, 3, 4, 5defgh, 6, 7, 8efgh, 9, 0,,, p #, 5, 6(solve each one, determine whether the ambiguous case should be considered), 7, 8, 0, a, p #, 3, 4, 5, 6, 7, 8, p #3ad, 4ac, 5, 6, 7, 8, 9,, 3, 6 Review p p #,, 3abdf, 4, 5, 6, 7, 8, 0(solve both triangles),, 3, 4, 5 # 6, 8abc, 0,, 3

2 MCR3U UNIT #6: TRIGONOMETRY Prerequisite Skills Trigonometric Ratios in Right Triangles * The 3 primar trig ratios in a right triangle are: sin θ = opposite hpotenuse cos θ = adjacent hpotenuse tan θ = opposite adjacent Remember : SOH CAH TOA B Hpotenuse Opposite A θ Adjacent C To solve a triangle means to determine all unknown side lengths and angle measures. To determine a trig ratio on our calculator, ou use the SIN, COS, or TAN function. To determine an angle measure on our calculator, ou use the SIN -, COS -, or TAN - function. Examples :. Calculate the following trig ratios, to 3 decimal places. a) sin54 b) cos39 c) tan70. Calculate the following angle measures, to the nearest degree. a) sina = 0.4 b) cosθ = 0.83 c) tanp = 4/3 3. Solve each triangle. Round side lengths to one decimal place and angle measures to the nearest degree. a) PQR with P = 43, Q = 90, and p = 5.5cm b) DEF with F = 90, d = 6.3cm, and f = 8.0cm

3 The Sine Law in Acute Triangles * The Sine Law : sina = sinb = sinc a b c A b c c C a B where A, B, and C are the angle measures and a, b, and c are the side lengths. * To use the Sine Law, ou need to be given: the measure of side lengths and non-contained angle, or the measure of angles and an side lengths. Examples : Solve each triangle. Round side lengths to one decimal place and angle measures to the nearest degree. a) ABC with A = 59, B = 76, and c = 5cm. b) GHI with H = 6, g = 8.5cm, and h = 9.cm. The Cosine Law in Acute Triangles A * The Cosine Law : a = b + c [bc(cosa)], or b = a + c [ac(cosb)], or c = a + b [ab(cosc)] b c C a B where A, B, and C are the angle measures and a, b, and c are the side lengths. * To use the Cosine Law, ou need to be given: the measure of side lengths and the contained angle, or the measure of all 3 side lengths. Examples : Solve each triangle. Round side lengths to one decimal place and angle measures to the nearest degree. a) ABC with A = 58, b = 0cm, and c = 6cm. b) PQR with p = 5.7cm, q = 5.0cm, and r = 4.cm.

4 4. SPECIAL ANGLES * An angle in standard position: The position of an angle rotating about the origin and its initial arm is the positive x-axis. Recall: The equation of a circle with its center at the origin is: x + = r The Unit Circle: center is at O(0, 0) radius is unit in length equation is : x + = Trig Ratios and the Unit Circle: OP is called the terminal arm, OA is the initial arm θ is the angle of rotation measured from the initial arm to the terminal arm. PON is a right triangle P P(x, ) r= θ O N A(, 0) x θ O x N opp sin θ = hp cos θ = adj hp opp tanθ = adj tanθ = x sinθ tanθ = cosθ sin θ = r sinθ = cos θ = x r x cosθ = sin θ = cos θ = x Therefore, the coordinates of an point (x, ) on a unit circle are related to θ such that (, ) = ( cosθ, sinθ ) x and sinθ tan θ = cosθ

5 Example: Determine the coordinates of the point on the unit circle for the following angles in standard position, to 3 decimal places. Draw the angles in standard position. a) θ = 40 b) θ = 30 c) θ = 5 d) θ = 340 As we rotate around the unit circle, the signs of the different trigonometric ratios change. Quadrant I : x is positive and is positive (x, ) Cosine is positive Sine is positive II I Tan is positive (-x, ) (x, ) Quadrant II: x is negative, is positive (-x, ) Cosine is negative Sine + ve All + ve Sine is positive Tan is negative Quadrant III: x is negative, is negative (-x, -) Cosine is negative Sine is negative Tan + ve Cos + ve Tan is positive Quadrant IV: x is positive, is negative (x, -) (-x, -) (x, -) Cosine is positive III IV Sine is negative Tan is negative We can remember the sign of each trigonometric function in each quadrant b using the CAST rule. C cosθ is positive in the 4 th quadrant A all trig ratios are positive in the st quadrant S A S sinθ is positive in the nd quadrant T tanθ is positive in the 3 rd quadrant T C Example: State whether each trig ratio is positive or negative without using a calculator. Then check with a calculator. a) sin0 b) cos05 c) tan340 d) cos- e) tan79

6 The Trigonometric Ratios of 0, 90, 80, 70, and 360. Angle 0 : Angle 90 : Angle 80 : Angle 70 : sin0 = sin90 = sin80 = sin70 = cos0 = cos90 = cos80 = cos70 = tan0 = tan90 = tan80 = tan70 = P(0,) r = P(-, 0) P(, 0) x P(0, -) Special Angles: 45, 30 and 60 When we use a calculator to determine trigonometric function values, we are approximating up to 9 decimal places. However, for some special angles, exact values can be determined from geometric relationships. A reference angle is the acute angle between the terminal arm and the x-axis. ) Angle 45 and its multiples (35, 5, 35 ) sin45 = cos45 = tan45 = 45 45

7 Use the CAST rule to determine the sign of the trigonometric ratio. Example: Draw each angle in standard position. Determine the EXACT VALUES of the primar trig ratios for each angle. a) θ = 35 b) θ = -45 c) θ = 5 ) Angle 30 (50, 0, 330 ) and 60 (0, 40, 300 ) and their multiples: 3 sin30 =, sin60 = 60 cos30 = 3, cos60 = 3 30 tan30 =, tan60 = 3 3 Examples: Draw each angle in standard position. Determine the EXACT VALUES of the primar trig ratios for each angle. a) θ = 50 b) θ = 40 c) θ = -30 d) θ = -300

8 4. COTERMINAL ANGLES AND RELATED ANGLES For an circle, we can determine the trigonometric ratio for θ. If (x, ) is an point on the terminal arm of a circle, the trig ratios can be determined as follows: sin θ = where r r = x + cos θ = x r tan θ = x r P(x, ) θ x Example : Given that 4 sin = 5 A and that A lies in the first quadrant, a) Determine the exact values for cosa and tana x b) Determine the primar trigonometric ratios for another angle between 0 and 360 that has the same sine value c) How are the two angles related? Use a calculator to determine the two angles, to the nearest degree.

9 Example : Given that cos 5 = 3 a) Determine the exact values for sina and tana. A and that A lies in the second quadrant, b) Determine the primar trigonometric ratios for another angle between 0 and 360 that has the same cosine value. x c) Use a calculator to determine the two angles between 0 and 360, to the nearest degree. Example 3: Given that 5 tan = 6 A and that A lies in the first quadrant, a) Determine the exact values for sina and cosa x b) Determine the primar trigonometric ratios for another angle between 0 and 360 that has the same tangent value c) Use a calculator to determine the two angles between 0 and 360, to the nearest degree.

10 Example 4: Without using a calculator, determine two angles between 0 and 360 that have a a) tangent of b) cosine of 3 3 c) sine of Example 5: The point P(3, -6) is on the terminal arm of A. a) Determine the primar trigonometric ratios for A. b) Determine the primar trigonometric ratios for B, such that B has the same cosine as A. c) Use a calculator and a diagram to determine the measures of A and B, to the nearest degree.

11 Direction of Rotation: When θ > 0, the rotation is counterclockwise When θ < 0, the rotation is clockwise. P P When P moves around the circle, the motion is repeated after P has rotated 360. So b adding or subtracting multiples of 360 to θ, we can determine other angles for which the position of P is the same. * Coterminal Angles: Angles in standard position that have the same terminal arm. To find an coterminal angle to θ: θ n, where n is an integer. Example 6: For each given angle: a) θ = 45 and b) θ = -0 (i) Draw the angle θ in standard position. (ii) Determine two positive angles coterminal with θ. (iii) Determine two negative angles coterminal with θ. (iv) Write an expression to represent an angle coterminal with θ.

12 4.3 RECIPROCAL TRIGONOMETRIC RATIOS * Reciprocals: two expressions that have a product of. (eg. 4 and ¼, x and /x) * The primar trigonometric ratios have reciprocal ratios. The reciprocals of sine, cosine and tangent are respectivel called the cosecant, secant, and cotangent and are abbreviated as csc, sec, and cot. The Reciprocal Trigonometric Ratios : For an right triangle, A hpotenuse csc θ = opposite csc θ = sinθ hpotenuse sec θ = adjacent sec θ = cosθ adjacent cot θ = opposite cot θ = tanθ opp hp C adj θ B * To use our calculators with the reciprocal trig ratios, we must use the reciprocal ke, or x x along with the sine, cosine, and tangent kes. Examples:. Determine the following trig ratios to 3 decimal places. a) csc36 b) sec80 c) cot5. Each angle is in the first quadrant. Determine the angle measure. Round answers to the nearest degree. a) cscθ =.564 b) secθ = 3.73 c) cotθ =.49

13 3. Calculate the 6 trig ratios for θ = 9, to 3 decimal places. 4. Write the 6 trig ratios for B. A 8cm 7cm C 5cm B 5. The point P(-5, -6) lies on the terminal arm of θ in standard position. a) Determine exact expressions for cscθ, secθ, and cotθ. b) Determine the measure of θ, to the nearest degree. 6. Angle θ is in the 4 th quadrant and tan 7 = 3 θ. a) Find the other 5 trig ratios, to 3 decimal places. b) Determine the smallest positive value of θ to the nearest degree.

14 7. Solve for both values of θ between 0 and 360. Round to the nearest degree. a) cscθ =.053 b) secθ =.49 c) cotθ = 0.88 d) cscθ = e) secθ = f) cotθ = Solve for all angles between 0 and 360 for the following reciprocal trigonometric ratios. Use a unit circle to help ou. a) cscθ = b) secθ = c) cotθ = 3 d) cscθ = undefined e) secθ = - 9. Determine the EXACT VALUES for the following reciprocal trigonometric ratios. a) csc0 b) cot5 c) csc405

15 4.4 PROBLEMS IN TWO DIMENSIONS * Recall : Given a right triangle, we use SOH CAH TOA or Primar Trig Ratios. Given side lengths and the contained angle, or all 3 side lengths, we use the Cosine Law. Given side lengths and of the non-contained angles, or angles and an side length, we use the Sine Law. We use the Sine Law and/or Cosine Law to solve an oblique triangle, acute or obtuse. * THE AMBIGUOUS CASE: A problem that has two or more solutions. When we are given side lengths and one of the non-contained angles, we use the Sine Law to solve the triangle. In this case, there ma be two possible solutions, one solution, or no solution. This is known as the ambiguous case. When there are possible triangles that can be constructed given these dimensions: an acute triangle and an obtuse triangle, we must solve both triangles. * To check for the Ambiguous Case: adj(sinθ) < opp < adj adjacent opposite θ adjacent If this statement is true, then there are triangles : one acute and one obtuse. Therefore, ou must solve both triangles. OR Use the Sine Law to find one of the missing angle measures. If ou can subtract this measure from 80 to get another angle, then there is another triangle to solve. * To solve both the acute and the obtuse : ) Use the Sine Law to find one of the missing angles (β). ) Solve the rest of the acute. 3) Use the angle ou found in step ) to determine the obtuse angle in the second triangle. (80 - β) 4) Solve the rest of the obtuse using the obtuse angle (80 - β) found in step 3) and the original dimensions.

16 Example : Determine whether or not the following triangles represent the ambiguous case. Solve both triangles, if necessar. Round all side lengths to one decimal place and angle measures to the nearest degree. a) PQR given p = 6.5cm, r = 8cm, and P = 38. b) ABC given a = 4cm, c = 5cm, and A = 46.

17 When solving word problems, it is still necessar to consider the possibilit of two triangles. Examples: Solve the following problems. It ma or ma not be necessar to consider the ambiguous case.. John works for a compan that cuts down dead trees for the cit. He needs to determine the height of the tree to ensure safet. He chooses a point 0 m from the base of the tree. From this point, the angle of elevation to the top of the tree is 46. Determine the height of the tree to the nearest tenth of a meter. 3. Three towns A, B, and C are located so that B is 5.0km awa from A. The angle at B is 55 and the angle at C is 73. Determine the distance between town B and C to the nearest km. 4. The radar screen at an air traffic control tower shows a Cherokee 5km from the tower in a direction 30 east of north, and a Skhawk 6km from the tower in a direction 40 east of north at their closest approach to each other. If the two aircrafts are less than km from each apart, the controller must file a report. Is a report necessar?

18 5. A acht is 9km from a sailboat and 8.5km from a lighthouse. From the sailboat, the acht and lighthouse are separated b an angle of 54. a) Is it necessar to consider the ambiguous case? b) Determine the distance from the sailboat to the lighthouse, to the nearest tenth of a kilometer.

19 4.5 PROBLEMS IN THREE DIMENSIONS Three dimensional problems involving triangles can be solved using one or more of the following trig tools: the Pthagorean Theorem, the primar trig ratios, the sine law and the cosine law. Examples: ) A radio antenna lies due north of Sam s house. Sam walks to Elena s house 00m awa 50 east of north. From Elena s house, the antenna is due west and the angle of inclination to the top of the radio tower is. Determine the height of the antenna, to the nearest metre. ) A surveor is on one side of a river. On the other side is a cliff of unknown height. To determine its height, the surveor las out a baseline AB of length 50m. From point A, she selects point C at the base of the cliff and measures CAB to be 5. She selects point D on the top of the cliff directl above point C and measures an angle of elevation of 3. She moves to point B and measures CBA as 6. Find the height of the cliff, to the nearest metre. D h C A 50m B

20 3) Justine is fling her hot-air balloon. She reports that her position is over a golf course located half-wa between Emertown and Fosterville at an altitude of 500m. Fosterville is 6km east of Danburg and Emertown is 6.5km from Danburg in a direction of 4 south of east. What is the angle of elevation of Justine s balloon as seen from Danburg, to the nearest degree? D F M E

21 4.6 TRIGONOMETRIC IDENTITIES Identit: An equation that is alwas true regardless of the value of the variable. A trigonometric identit is a relation among trig ratios that is true for all angles for which both sides are defined. The Basic Trig Identities: The Pthagorean Trig Identit : sin θ + cos θ = The Quotient Trig Identities : sinθ θ = cos θ tan and cos θ cot θ = sinθ The Reciprocal Trig Identities : θ = sin θ csc and θ = cos θ sec and cot θ = tan θ * We can use these basic trig identities to prove more complex trig identities. * We also use fraction skills, algebra skills (ex. Distributive Law), and factoring to help prove trig identities. * Identities can be used to simplif solutions to problems that result in trigonometric expressions. * To prove an identit, we show that the left side of the equation equals the right side of the equation. Examples : Prove the following identities. a) sin θ sin θ = tan θ b) tan θ + = tanθ sinθ cosθ

22 cos sin = cos sin d) + tan θ = sec θ 4 4 c) θ θ θ θ e) θ sin θ + cos θ = sec θ tan f) + sinθ cosθ = cosθ sinθ

Ch. 2 Trigonometry Notes

Ch. 2 Trigonometry Notes First Name: Last Name: Block: Ch. Trigonometry Notes.0 PRE-REQUISITES: SOLVING RIGHT TRIANGLES.1 ANGLES IN STANDARD POSITION 6 Ch..1 HW: p. 83 #1,, 4, 5, 7, 9, 10, 8. - TRIGONOMETRIC FUNCTIONS OF AN ANGLE