Trig Identities, Solving Trig Equations Answer Section

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1 Trig Identities, Solving Trig Equations Answer Section MULTIPLE CHOICE. ANS: B PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas. ANS: A PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas. ANS: C PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas 4. ANS: A PTS: REF: Application OBJ: 7. - Compound Angle Formulas 5. ANS: A PTS: REF: Application OBJ: 7. - Compound Angle Formulas 6. ANS: B PTS: REF: Application OBJ: 7. - Compound Angle Formulas 7. ANS: B PTS: REF: Application OBJ: 7. - Double Angle Formulas 8. ANS: D PTS: REF: Application OBJ: 7. - Double Angle Formulas 9. ANS: C PTS: REF: Knowledge and Understanding OBJ: Proving Trigonometric Identities 0. ANS: B PTS: REF: Application OBJ: Proving Trigonometric Identities. ANS: C PTS: REF: Knowledge and Understanding OBJ: Proving Trigonometric Identities. ANS: A PTS: REF: Knowledge and Understanding. ANS: B PTS: REF: Thinking 4. ANS: A PTS: REF: Knowledge and Understanding 5. ANS: A PTS: REF: Knowledge and Understanding 6. ANS: B PTS: REF: Knowledge and Understanding 7. ANS: D PTS: REF: Knowledge and Understanding 8. ANS: A PTS: REF: Knowledge and Understanding 9. ANS: A PTS: REF: Application 0. ANS: B PTS: REF: Application SHORT ANSWER. ANS: PTS: REF: Application OBJ: 7. - Compound Angle Formulas. ANS: Because the problem can be simplified to tan90 by using the compound angle formula backwards. PTS: REF: Communication OBJ: 7. - Compound Angle Formulas. ANS: PTS: REF: Thinking OBJ: 7. - Compound Angle Formulas 4. ANS: cos(8x) PTS: REF: Knowledge and Understanding OBJ: 7. - Double Angle Formulas

2 5. ANS: PTS: REF: Thinking OBJ: 7. - Double Angle Formulas 6. ANS: PTS: REF: Thinking OBJ: 7. - Double Angle Formulas 7. ANS: 7 5 PTS: REF: Thinking OBJ: 7. - Double Angle Formulas 8. ANS: 0 9 PTS: REF: Thinking OBJ: 7. - Double Angle Formulas 9. ANS: PTS: REF: Thinking OBJ: 7. - Double Angle Formulas 0. ANS: 5 PTS: REF: Thinking OBJ: 7. - Double Angle Formulas. ANS: sinθ PTS: REF: Thinking OBJ: Proving Trigonometric Identities. ANS: sec θ sec θ PTS: REF: Application OBJ: Proving Trigonometric Identities. ANS: π 6, 5π 6 PTS: REF: Thinking

3 4. ANS: 44.4, 5.57 PTS: REF: Thinking 5. ANS: 0 hours; noon PTS: REF: Thinking 6. ANS: II and III PTS: REF: Knowledge and Understanding 7. ANS: 60 PTS: REF: Knowledge and Understanding 8. ANS: π, 7π 6, π 6 PTS: REF: Thinking 9. ANS: 70.5, 99.59, 60.4, PTS: REF: Thinking 40. ANS: π PTS: REF: Thinking PROBLEM 4. ANS: sin π = sin π sin( π + π ) = Ê sin π ˆ Á cos Ê π ˆ Á + sin Ê π ˆ Á cos Ê π ˆ Á = Ê ˆ Á = Ê ˆ 4 Á = = PTS: REF: Application OBJ: 7. - Compound Angle Formulas 4. ANS: sin(π x) = sinx sinπ cosx sinx cosπ = sinx (0) cos x (sinx)() = sinx sinx = sinx PTS: REF: Application OBJ: 7. - Compound Angle Formulas

4 4. ANS: a) 65 b) 6 65 c) The ratios are determined using a compound angle formula and the component ratios, without any statement of the angle measurements. PTS: REF: Thinking OBJ: 7. - Compound Angle Formulas 44. ANS: a) It doesn t matter because the double angle formula needed is sinθ = sinθ cosθ, and the commutative law of multiplication allows for either number to be sine/cosine. b) i) sinx = 7 5 sinx = = 6 65 ii) cosx = 7 5 sinx = = 6 65 PTS: REF: Communication OBJ: 7. - Double Angle Formulas 45. ANS: She must develop a formula for cos x cosx = cos x cosx = cos x cosx + = cos x cos x =± cosx + cos π π cos 6 =± 6 + cos π =± + cos π =± + in terms of cosx. PTS: REF: Thinking OBJ: 7. - Double Angle Formulas 46. ANS: sin(a + b) = sina cosb + cosa sinb a = x b = x sin(x + x) = sinx cosx + cosx sinx sin(x) = sinx cosx PTS: REF: Communication OBJ: 7. - Double Angle Formulas 4

5 47. ANS: tan x + tan x = cosx tan x sec x = cosx cos x( tan x) = cosx cos x cos x tan x = cosx cos x cos x sin x cos x = cosx cos x sin x = cosx cosx = cosx PTS: REF: Thinking OBJ: Proving Trigonometric Identities 48. ANS: sinx + sinx cot x = csc x sinx( + cot x) = csc x sinx csc x = csc x sinx sin x = csc x sinx = csc x csc x = csc x PTS: REF: Application OBJ: Proving Trigonometric Identities 49. ANS: a) b) II, IV c) π 4 d) π 4, 7π 4 PTS: REF: Application 50. ANS: a) b) I, IV c) π 6 d) π 6, 5π 6 PTS: REF: Application 5. ANS: a) b) II, III c) d) 4.4, 5.57 PTS: REF: Application 5

6 5. ANS: a) centimetres; on the /0 (the top of the clock) b) Hour: Change the coefficient to a to shrink the highest and lowest points to and -. Multiply 0 by 600 to get (600 is the number of seconds in an hour) to stretch the equation horizontally Minute: Change the coefficient to a 5 to stretch the highest and lowest points to 4 and -4. Multiply 0 by 60 to get (60 is the number of seconds in a minute) to stretch the equation horizontally. 800 c) Change cosine to sine in each equation. It works because sine and cosine represent the relationship of the sides of a triangle with the tip of the hands as the angle of interest. PTS: REF: Communication 5. ANS: a) 5cos x + 4cosx = 0 Ê b) cosx ˆ x + ) = 0 Á 5 (cos c) 78.46, 80, 8.54 PTS: REF: Thinking 54. ANS: a).78 metres b) Twice. c) 0.5 seconds,.6 seconds, 4.7 seconds PTS: REF: Application 55. ANS: a) times b).96 cm c) 0.6 seconds, 0.6 seconds PTS: REF: Application 6

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