Rucete ✏ AP Precalculus In a Nutshell
8. Trigonometric Functions — Practice Questions 2
This chapter covers unit circle definitions, graph transformations, trigonometric identities, inverse functions, equations, and sinusoidal modeling essential for AP, SAT, and university-level exams.
(Multiple Choice — Click to Reveal Answer)
1. Convert −330° to radians.
(A) −11π/6
(B) −5π/6
(C) 11π/6
(D) −5π/3
Answer
(A) — −330° × (π/180) = −11π/6.
2. Which coordinate on the unit circle corresponds to angle θ?
(A) (sin θ, cos θ)
(B) (cos θ, sin θ)
(C) (tan θ, cot θ)
(D) (sec θ, csc θ)
Answer
(B) — The definition of the unit circle.
3. Evaluate cos(−π/3).
(A) −1/2
(B) 1/2
(C) √3/2
(D) −√3/2
Answer
(B) — Cosine is an even function: cos(−x) = cos x.
4. What is the period of y = sin(πx)?
(A) 2
(B) π
(C) 1
(D) 4
Answer
(A) — Period = 2π / π = 2.
5. Which function has amplitude 4 and midline y = −2?
(A) y = 4sin x − 2
(B) y = −4sin x + 2
(C) y = 2sin x − 4
(D) y = 4sin(x + 2)
Answer
(A) — Amplitude is |4|; midline is −2.
6. Evaluate sin(5π/6).
(A) −1/2
(B) √3/2
(C) 1/2
(D) −√3/2
Answer
(C) — Reference angle is π/6, and sine is positive in QII.
7. If tan θ = −1 and θ is in Quadrant IV, what is θ?
(A) π/4
(B) 3π/4
(C) 7π/4
(D) 5π/4
Answer
(C) — tan is negative in QIV, and reference angle is π/4.
8. Which identity is always true?
(A) 1 + tan²θ = sec²θ
(B) sin²θ − cos²θ = 1
(C) sec²θ − tan²θ = −1
(D) sin²θ + cos²θ = sec²θ
Answer
(A) — This is a fundamental Pythagorean identity.
9. Convert (3π/4) radians to degrees.
(A) 90°
(B) 120°
(C) 135°
(D) 150°
Answer
(C) — (3π/4) × (180/π) = 135°.
10. What is the period of y = cos(2x)?
(A) π/2
(B) π
(C) 2π
(D) 4π
Answer
(B) — Period = 2π / 2 = π.
11. Evaluate csc(π/2).
(A) 0
(B) 1
(C) −1
(D) Undefined
Answer
(B) — csc = 1/sin; sin(π/2) = 1.
12. A left horizontal shift of π/6 is represented by:
(A) y = sin(x − π/6)
(B) y = sin(x + π/6)
(C) y = sin(π/6 − x)
(D) y = sin(π − x)
Answer
(B) — x + π/6 shifts left.
13. If cos θ = 12/13 and θ is in Quadrant IV, find sin θ.
(A) 5/13
(B) −5/13
(C) 13/5
(D) −13/5
Answer
(B) — sin = −√(1 − cos²θ) = −5/13.
14. Evaluate sin(−π/2).
(A) −1
(B) 0
(C) 1
(D) Undefined
Answer
(A) — Sine is odd: sin(−θ) = −sin θ.
15. Which parameter changes the period of y = A sin(Bx) + D?
(A) A
(B) B
(C) D
(D) A and D
Answer
(B) — Period = 2π / |B|.
16. What is the value of sin²(π/4) + cos²(π/4)?
(A) 0
(B) 1/2
(C) 1
(D) √2/2
Answer
(C) — Pythagorean identity always equals 1.
17. If sin θ = 2/3 and θ is in Quadrant II, what is tan θ?
(A) 2/√5
(B) −2/√5
(C) √5/2
(D) −√5/2
Answer
(B) — cos θ = −√(1 − 4/9) = −√5/3; tan = sin/cos = (2/3)/(−√5/3) = −2/√5.
18. Which identity is true?
(A) sin(π − x) = sin x
(B) cos(π − x) = cos x
(C) tan(π − x) = tan x
(D) sin(π + x) = sin x
Answer
(A) — Sine is positive in QII: sin(π − x) = sin x.
19. Evaluate tan(−π/4).
(A) 1
(B) −1
(C) 0
(D) Undefined
Answer
(B) — Tangent is odd: tan(−θ) = −tan θ.
20. What is the range of y = 3 − 2cos x?
(A) [1, 5]
(B) [−1, 5]
(C) [−5, 0]
(D) [0, 3]
Answer
(A) — cos x ∈ [−1, 1] ⇒ output ∈ [3 − 2(1), 3 − 2(−1)] = [1, 5].
21. Convert 7π/12 to degrees.
(A) 60°
(B) 90°
(C) 105°
(D) 120°
Answer
(C) — 7π/12 × (180/π) = 105°.
22. Which angles satisfy cos x = −√2/2?
(A) π/4, 7π/4
(B) 3π/4, 5π/4
(C) π/2, 3π/2
(D) 5π/6, 7π/6
Answer
(B) — Cosine is negative in Quadrants II and III.
23. Evaluate sec(π/3).
(A) 1/2
(B) √3
(C) 2
(D) −2
Answer
(C) — sec = 1/cos; cos(π/3) = 1/2.
24. What is the phase shift of y = 3sin(2x − π)?
(A) π/2 left
(B) π/2 right
(C) π right
(D) π/2 upward
Answer
(B) — 2x − π = 2(x − π/2), so shift right π/2.
25. Which has amplitude 3, period 4π, and midline y = −1?
(A) y = 3sin(x/2) − 1
(B) y = 3sin(x/4) − 1
(C) y = 3sin(2x) − 1
(D) y = 3sin(x) − 1
Answer
(B) — Period = 2π/B = 4π ⇒ B = 1/2; amplitude = 3; shift = −1.
26. Which identity correctly represents sin(75°)?
(A) sin(45° + 30°) = sin45 cos30 + cos45 sin30
(B) sin(45° + 30°) = sin45 sin30 + cos45 cos30
(C) sin(45° − 30°) = sin45 cos30 − cos45 sin30
(D) sin(30° − 45°) = sin30 cos45 − cos30 sin45
Answer
(A) — Sum identity for sine.
27. Which identity converts product to sum?
(A) sin A cos B = (1/2)[sin(A + B) + sin(A − B)]
(B) cos A cos B = (1/2)[sin(A + B) + sin(A − B)]
(C) sin A sin B = (1/2)[cos(A − B) − cos(A + B)]
(D) sin A cos B = (1/2)[cos(A + B) + cos(A − B)]
Answer
(A) — Standard product-to-sum identity.
28. Solve 2sin x + √3 = 0 for x ∈ [0, 2π].
(A) 5π/6, 7π/6
(B) 2π/3, 4π/3
(C) 7π/6, 11π/6
(D) π/3, 2π/3
Answer
(C) — sin x = −√3/2 → x = 210°, 330°.
29. If tan x = 5/12 and x is in Quadrant I, what is sin(2x)?
(A) 60/169
(B) 120/169
(C) 5/13
(D) 12/13
Answer
(B) — sin(2x) = 2tan/(1+tan²) = 2·(5/12)/(1+25/144) = 120/169.
30. What is the domain and range of y = arctan(x)?
(A) Domain (−∞, ∞), Range (−π/2, π/2)
(B) Domain (0, ∞), Range (−π/2, π/2)
(C) Domain (−π/2, π/2), Range (−∞, ∞)
(D) Domain (−∞, ∞), Range [−π/2, π/2]
Answer
(A) — Principal branch excludes ±π/2.
31. Which is a double-angle identity for cosine in terms of sine only?
(A) cos(2x) = 1 − 2sin²x
(B) cos(2x) = 2cos²x − 1
(C) cos(2x) = cos²x − sin²x
(D) cos(2x) = 1 + 2sin²x
Answer
(A) — Standard identity.
32. Solve for θ ∈ [0, 2π]: 2cos²θ − 3cosθ + 1 = 0.
(A) θ = 0, π/3, 5π/3
(B) θ = π/2, 3π/2
(C) θ = π/3, π, 5π/3
(D) θ = 0, π, 2π
Answer
(A) — cosθ = 1 or 1/2 ⇒ θ = 0, π/3, 5π/3.
33. Evaluate sin(arccos(4/5)).
(A) 3/5
(B) 4/5
(C) −3/5
(D) −4/5
Answer
(A) — Right triangle: adjacent=4, hypotenuse=5 ⇒ opposite=3.
34. Solve for x in (−π, π): tan(x + π/6) = 0.
(A) x = −π/6
(B) x = −π/6, 5π/6
(C) x = π/6 only
(D) x = π/3 only
Answer
(B) — x + π/6 = kπ ⇒ x = −π/6 + kπ; valid solutions are −π/6, 5π/6.
35. Expand using sum identity: cos(75°).
(A) cos(45° + 30°) = cos45 cos30 − sin45 sin30
(B) cos(45° + 30°) = cos45 cos30 + sin45 sin30
(C) cos(30° − 45°) = cos30 cos45 + sin30 sin45
(D) cos(45° − 30°) = cos45 cos30 + sin45 sin30
Answer
(A) — Cos(a + b) = cos a cos b − sin a sin b.
36. Solve for x in [0, 2π]: 3sin x − 2 = 0.
Answer
x = sin⁻¹(2/3), π − sin⁻¹(2/3) — sin x = 2/3 has two solutions in [0, 2π].
37. Find the exact value: sin(π/12).
Answer
(√6 − √2)/4 — Use sin(45° − 30°) = sin45 cos30 − cos45 sin30.
38. For y = −3 cos(4x + π) + 2, state amplitude, period, phase shift, and midline.
Answer
Amplitude 3; Period π/2; Phase shift left π/4; Midline y = 2 — 4x + π = 4(x + π/4).
39. Solve for x ∈ [0, 2π]: tan x = √3/3.
Answer
x = π/6, 7π/6 — Reference angle π/6; tangent positive in QI and QIII.
40. Prove the identity: 1 − cos(2x) = 2sin²x.
Answer
True — From cos(2x) = 1 − 2sin²x ⇒ 1 − cos(2x) = 2sin²x.
41. Solve for x ∈ [0, 2π]: cos(2x) = −1/2.
Answer
x = π/3, 2π/3, 4π/3, 5π/3 — 2x = 2π/3, 4π/3 (mod 2π), divide by 2.
42. Evaluate exactly: tan(105°).
Answer
−(2 + √3) — tan(60+45) = (√3 + 1)/(1 − √3) = −(2 + √3).
43. Find the principal value: arctan(−1).
Answer
−π/4 — Range (−π/2, π/2).
44. Solve for x ∈ [0, 2π]: 2sin²x − sin x − 1 = 0.
Answer
x = π/6, 5π/6, 3π/2 — Factor (2s − 1)(s + 1) = 0 ⇒ s = 1/2 or −1.
45. Write as a single trig function: sin x cos 2x + cos x sin 2x.
Answer
sin(3x) — Sine addition: sin(a + b) = sin a cos b + cos a sin b (a = x, b = 2x).
46. For y(t) = 2 + 5 sin((π/3)t − π/6), give amplitude, period, and phase shift.
Answer
Amplitude 5; Period 6; Phase shift right 1/2 — (π/3)t − π/6 = (π/3)(t − 1/2).
47. Find the exact value: cos(255°).
Answer
(−√6 + √2)/4 — 255° = 180° + 75° ⇒ cos(255) = −cos(75) = −(√6 − √2)/4.
48. Solve for θ ∈ [0, 2π]: sec θ = −2.
Answer
θ = 2π/3, 4π/3 — cos θ = −1/2 in QII and QIII.
49. State the domain of y = tan(3x − π/2).
Answer
All real x except x = (π + kπ)/3, k ∈ ℤ — tan undefined when 3x − π/2 = π/2 + kπ ⇒ 3x = π + kπ.
50. The tide height model is H(t) = 1.2 + 0.5 sin((2π/24)t − π/3) (t in hours). When is the first maximum after t = 0?
Answer
t = 10 hours — Max when argument = π/2: (2π/24)t − π/3 = π/2 ⇒ (π/12)t = 5π/6 ⇒ t = 10.