Related Angles WS # 6.1. Co-Related Angles WS # 6.2. Solving Linear Trigonometric Equations Linear

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1 UNIT 6 TRIGONOMETRIC IDENTITIES AND EQUATIONS Date Lesson Text TOPIC Homework (60) 7.1 Related Angles WS # (61) 7.1 Co-Related Angles WS # (63) 7. Compound Angle Formulas I Sine and Cosine Pg. 400 # 1, b, 3, 4abde, 5abdf, 6abce, 8acd, 9abcd, (64) 7. Compound Angle Formulas II Tangent Pg. 400 # a, 4cf, 5ce, 6df, 8bef, 9ef, 10, (65) 7.3 Double Angle Formulas Pg. 407 # 1,, 3, 5, 8, 10, 11, 1ac, 13cd (69) 7.5 Solving Linear Trigonometric Equations Linear Pg. 47 # 10doso, 11, 1 WS 6.6 # 1, 6.7 (70) 7.6 Solving Quadratic Trigonometric Equations Quadratic Pg. 437 # 13, 14, 15 WS 6.6 # (3 6)doso (71) 6.9 (7) Review for Unit 6 Test Pg. 440 # 1 6, UNIT 6 TEST on Extra Review Pg. 441 # (66) 6.10 (67) 6.10 (68) Proving Trigonometric Identities using the reciprocal, quotient, and Pythagorean identities Add/Sub formulas Proving Trigonometric Identities using related, co related angles, Add/Sub formulas and double angle formulas Proving Trigonometric Identities using a variety of formulas and identities WS 6.10 Part A WS 6.10 Part B WS 6.10 Part C Pg 440 # 7-9 Dec. 1 TRIG IDENTITIES QUIZ

2 MHF 4U Lesson 6.1 Related Angle Identities sin (π ) = sin = cos (π ) = cos = tan (π ) = tan = 0 r / sin (π + ) = sin (π ) = sin ( ) = cos (π + ) = cos (π ) = cos ( ) = tan (π + ) = tan (π ) = tan ( ) = 3 Ex. 1 Express each of the following as a function of its related acute angle and find its exact value. a) sin 3 b) tan c) sec 6

3 11 d) csc 6 4 e) cot 3 cot g) cos 45 f) 10 WS 6.1

4 MHF 4U Lesson 6. Co-Related Angle Identities sin sin cos cos tan tan 0 r / 3 sin 3 sin 3 cos 3 cos 3 tan 3 tan Ex. 1 Express each of the following as a function of its co-related acute angle and find its exact value. 3 4 a) cos 3 b) tan 300 c) cot 3

5 Ex. Prove: cos sin Ex. 3 Simplify: sin x sin x sin x sin x Ex. 4 Express each of the following as a function of its co-elated acute angle ad find its exact value. a) 5 sin b) 6 4 tan 3 Ex. 5 Express as a function of its co-related acute angle. a) 3 csc b) sec WS 6.

6 MHF 4U Lesson 6.3 Addition and Subtraction Formulas for Sine and Cosine B D 90 - E + O C F O is centre of unit circle Unit Circle Addition Formula for Sine sin (a + b) = sin a cos b + cos a sin b

7 Subtraction Formula for Sine sin (a b) = sin a cos b cos a sin b PROOF: sin (a b) Addition Formula for Cosine cos (a + b) = cos a cos b sin a sin b PROOF: cos (a + b) Subtraction Formula for Cosine cos (a b) = cos a cos b + sin a sin b PROOF: cos (a b)

8 Ex. 1 Find the exact value of sin 1. Ex. If sin a = 4 5, a 3 and cos b = 5 13, b, evaluate cos(a - b). Ex. 3 Prove that cos(π + x) = cos x Ex. 4 Express sin 3x cos 4x cos 3x sin 4x as a single trigonometric function. Ex. 5 Find the exact value of cos 70 cos 40 + sin 70 sin 40 Pg. 400 # 1, b, 3, 4abde, 5abdf, 6abce, 8acd, 9abcd, 1

9 MHF 4U Lesson 6.4 Trigonometric Addition and Subtraction Formulas for Tangent tan (a + b) = PROOF: tan (a + b) = Addition Formula for Tangent tan a tan b 1 tan a tanb Subtraction Formula for Tangent tan a tan b tan (a b) = 1 tan a tanb PROOF: tan (a b) =

10 Ex. If sin a = 4 5, a 3 and cos b = 5 13, b evaluate tan (a + b) Ex. Find the exact value of 19 tan. 1 Pg. 400 # a, 4cf, 5ce, 6df, 8bef, 9ef, 10, 11

11 MHF 4U Lesson 6.5 Double Angle Formulas Double Angle Formula for Sine sin x = sin x cos x PROOF: Double Angle Formulas for Cosine cos x = cos x sin x = cos x 1 = 1 sin x PROOF: Double Angle Formula for Tangent tanx tanx 1 tan x PROOF:

12 Ex. 1 If cos a =, find the value of cos 4a. 3 Ex. Evaluate sin 8. Ex. 3 If tan x = 4 3, x 3, find the value of tan x. Pg. 407 # 1,, 3, 5, 8, 10, 11, 1ac,13cd

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14 MHF 4U Lesson 6.6 Solving Linear Trigonometric Equations Strategies use known identities and formulas to express the equation in terms of a single trigonometric function use algebraic methods to solve for the trig function use special triangles, trig graphs (calculator only if necessary ie: cos x = 0.974) to find the RAA and to solve for the angle within the given domain solve for the variable check your answer Ex. Solve each of the following. a) sin x cos x cos x, 0 x b) sin x tan x 0, 0 x sin x c) 3 0, 0 x 360 cos x d) 4sin x 4 sin x 6, 0 x correct to d. p. Pg. 47 # 10doso, 11, 1 WS 6.7 # (1, )doso

15 MHF 4U Lesson 6.7 Solving Quadratic Trigonometric Equations Ex. Solve each of the following. a) cos x cos x sin x + 3 = 0, 0 x b) csc (x) 1 = 0, 0 x 1 c) sin x - 7sin x + 3, - x d) sec x 0, 0 x 360 cos x

16 MHF 4U Lesson 6.10 Proving Trigonometric Identities Strategies begin with the more complex side and continue until it is the same as the simpler side sometimes it is easier to work with both sides and manipulate them until they are the same express all functions in terms of sine and/or cosine (if all terms are in terms of the same trig function, it is often easier to work with those instead of changing to sin and cos) look for factoring opportunities and look for conjugates often just doing the operations in the identity will help immensely look for squares and use the Pythagorean identities express all functions with the same argument *** this is not an exhaustive list of strategies Ex. Prove each of the following. sin x a) 1 cos x b) 1 cos x tan A sin A 1 tan A

17 4 4 c) cos x sin x cos x d) sin 4x 8cos x sin x 4cos x sin x 3 e) cos( x y) cos( x y) cos x cos y 1 sin x f) csc x tan x cos x

18 x 1 tan g) cos x h) x 1 tan 3tan x tan x tan 3x 1 3tan x 3 Day 1: WS 6.10 Part A Day : WS 6.10 Part B Day 3: WS 6.10 Part C