Rucete ✏ AP Precalculus In a Nutshell
8. Trigonometric Functions — Practice Questions 3
This chapter explores real-world modeling, inverse trigonometric functions, transformations, identities, equations, phase shifts, and calculus-related interpretations of trigonometric functions.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following best describes the graph of y = sin(x + π/3)?
(A) A sine wave shifted right by π/3
(B) A sine wave shifted left by π/3
(C) A cosine wave shifted right by π/3
(D) A cosine wave shifted left by π/3
Answer
(B) — Inside + causes a horizontal shift left.
2. A Ferris wheel is modeled by H(t) = 40 + 35 sin( (π/6)t − π/2 ), where t is minutes. What is the maximum height?
(A) 35
(B) 40
(C) 75
(D) 5
Answer
(C) — Midline 40 + amplitude 35 = max height 75.
3. Which angle lies in Quadrant III?
(A) 2π/3
(B) 5π/4
(C) π/6
(D) 7π/4
Answer
(B) — 5π/4 > π and < 3π/2.
4. A trigonometric function has amplitude 3, period π, and midline y = −2. Which could be its equation?
(A) y = 3cos(2x) − 2
(B) y = 3cos(x/2) − 2
(C) y = −3sin(2x) + 2
(D) y = 3sin(x) + 2
Answer
(A) — Period = 2π/B = π ⇒ B = 2.
5. Which of the following is equivalent to tan(π − x)?
(A) tan x
(B) −tan x
(C) cot x
(D) −cot x
Answer
(B) — Tangent is periodic with a sign change in Quadrant II.
6. Suppose sin θ = 0.8 and θ is acute. What is cos θ?
(A) 0.2
(B) 0.6
(C) 0.8
(D) √(1 − 0.8)
Answer
(B) — cos = √(1 − 0.64) = 0.6.
7. Which of the following has a period of π/2?
(A) y = sin 4x
(B) y = sin(x/2)
(C) y = tan 2x
(D) y = cos x
Answer
(A) — 2π/4 = π/2.
8. What is the range of y = 2 − 5cos(x)?
(A) [−5, 2]
(B) [−3, 7]
(C) [−7, 3]
(D) [−3, 5]
Answer
(B) — cos ∈ [−1,1], so 2 − 5cos(x) ∈ [2 − 5(1), 2 − 5(−1)] = [−3, 7].
9. Which of the following angles corresponds to sin θ = −1?
(A) θ = 3π/2
(B) θ = π/2
(C) θ = π
(D) θ = 0
Answer
(A) — sin is −1 at 270°.
10. If a trigonometric function has a maximum at x = π/6 and a minimum at x = 7π/6, what is its period?
(A) π/3
(B) π
(C) 2π/3
(D) 4π/3
Answer
(B) — From max to next max is one period; midpoint between is π/6 to 7π/6 is π ⇒ full period = 2π.
11. Which identity is true?
(A) sec θ = 1/cos θ
(B) sec θ = 1/sin θ
(C) sec θ = cos θ
(D) sec θ = tan θ
Answer
(A) — Definition of secant.
12. Which point lies on the unit circle?
(A) (1.2, 0.5)
(B) (−√3/2, 1/2)
(C) (−1, −1)
(D) (0.8, 0.8)
Answer
(B) — (−0.866, 0.5) gives x² + y² = 1.
13. What is the period of y = tan(3x)?
(A) π
(B) π/3
(C) 3π
(D) 2π/3
Answer
(B) — Period = π/3.
14. Which angle is coterminal with 7π/6?
(A) π/6
(B) 13π/6
(C) −5π/6
(D) π/3
Answer
(B) — 7π/6 + 2π = 7π/6 + 12π/6 = 19π/6 (simplify to 13π/6 by subtracting 2π).
15. Which expression is equivalent to sin²x + cos²x?
(A) tan²x
(B) 1
(C) sec²x
(D) 0
Answer
(B) — Pythagorean identity.
16. A sinusoidal function has midline y = 4 and maximum value y = 10. What is its amplitude?
(A) 4
(B) 6
(C) 10
(D) 14
Answer
(B) — Amplitude = max − midline = 10 − 4 = 6.
17. Which of the following represents a vertical reflection of y = sin(x) across the x-axis?
(A) y = sin(−x)
(B) y = −sin(x)
(C) y = sin(x) + 1
(D) y = cos(x)
Answer
(B) — Multiplying by −1 reflects over the x-axis.
18. If cos x = −3/5 and x is in Quadrant III, what is sin x?
(A) 4/5
(B) −4/5
(C) 3/4
(D) −3/4
Answer
(B) — sin is negative in QIII; sin = −√(1 − 9/25) = −4/5.
19. Which identity is correct?
(A) sin(2x) = sin²x − cos²x
(B) sin(2x) = 2sin x cos x
(C) cos(2x) = sin x + cos x
(D) tan(2x) = tan x
Answer
(B) — This is the double-angle identity for sine.
20. What is the phase shift of y = cos(2x + π/2)?
(A) π/2 left
(B) π/2 right
(C) π/4 left
(D) π/4 right
Answer
(A) — 2x + π/2 = 2(x + π/4), which is a left shift of π/4, not π/2. Correct shift is π/4 left.
20. What is the phase shift of y = cos(2x + π/2)?
(A) π/4 left
(B) π/4 right
(C) π/2 left
(D) π/2 right
Answer
(A) — Factor inside: 2x + π/2 = 2(x + π/4) ⇒ shift left by π/4.
21. Evaluate sin(π + x) in terms of x.
(A) sin x
(B) −sin x
(C) cos x
(D) −cos x
Answer
(B) — Sine shifts by π produce sign reversal.
22. Which graph has a period of 120°?
(A) y = sin(2x)
(B) y = sin(3x)
(C) y = sin(πx/60)
(D) y = sin(60x)
Answer
(C) — Period in degrees is 360°/(π/60 × 180/π) = 120°.
23. A sound wave is modeled by P(t) = 5cos(4πt) + 3. What is the midline?
(A) y = 0
(B) y = 3
(C) y = 5
(D) y = 8
Answer
(B) — The constant term indicates midline.
24. What is the solution to cos x = 0 in the interval [0, 2π]?
(A) π/2 only
(B) 3π/2 only
(C) π/2, 3π/2
(D) 0, π
Answer
(C) — Cosine equals zero at 90° and 270°.
25. Which is equivalent to tan θ?
(A) sin θ / cos θ
(B) cos θ / sin θ
(C) 1 / sin θ
(D) 1 / cos θ
Answer
(A) — Definition of tangent.
26. Solve for x in [0, 2π]: sin(2x) = √2/2.
(A) π/8, 7π/8
(B) π/4, 3π/4, 5π/4, 7π/4
(C) π/8, 3π/8, 9π/8, 11π/8
(D) π/6, 5π/6
Answer
(C) — 2x = π/4, 3π/4, 5π/4, 7π/4 ⇒ x = π/8, 3π/8, 9π/8, 11π/8.
27. Which of the following is the correct domain of y = arccos(x)?
(A) (−∞, ∞)
(B) [−1, 1]
(C) (0, 1)
(D) [0, π]
Answer
(B) — Arccos is defined only for −1 ≤ x ≤ 1.
28. Solve the equation cos(x) + 1 = 0 in [0, 2π].
(A) x = 0
(B) x = π
(C) x = π/2
(D) x = 2π
Answer
(B) — cos(x) = −1 at x = π only.
29. Which identity is correct?
(A) cos(a + b) = cos a cos b − sin a sin b
(B) cos(a + b) = cos a cos b + sin a sin b
(C) cos(a − b) = cos a sin b + sin a cos b
(D) cos(a − b) = sin a sin b − cos a cos b
Answer
(A) — Standard angle sum identity.
30. Solve for x in [0, 2π]: tan(x) = −√3.
(A) 2π/3, 5π/3
(B) π/6, 7π/6
(C) 5π/6, 11π/6
(D) 3π/4, 7π/4
Answer
(A) — tan is negative in Quadrants II and IV; reference angle π/3.
31. If y = arctan(x), which of the following is true?
(A) tan(y) = x
(B) sin(y) = x
(C) cos(y) = x
(D) y = 1/x
Answer
(A) — By definition of inverse function.
32. What is the amplitude of y = −2cos(3x − π/2) + 5?
(A) 2
(B) −2
(C) 3
(D) 5
Answer
(A) — Amplitude is |−2| = 2.
33. If cos(x) = 4/5 and sin(x) > 0, which quadrant is x in?
(A) Quadrant I
(B) Quadrant II
(C) Quadrant III
(D) Quadrant IV
Answer
(A) — Both sine and cosine positive.
34. Solve for x in [0, 2π]: 4cos²x − 3 = 0.
(A) x = π/6, 11π/6
(B) x = π/3, 5π/3
(C) x = π/4, 3π/4
(D) x = π/2, 3π/2
Answer
(B) — cos²x = 3/4 ⇒ cos x = ±√3/2 ⇒ x = π/6, 5π/6, 7π/6, 11π/6.
35. Which expression simplifies to 1?
(A) sin²θ + cos²θ
(B) tan²θ − sec²θ
(C) sec²θ − 1
(D) 1 − tan²θ
Answer
(A) — Pythagorean identity.
36. Solve for x in [0, 2π]: 3cos x − 1 = 0.
Answer
x = cos⁻¹(1/3), 2π − cos⁻¹(1/3) — cos x = 1/3 has two valid solutions in Quadrants I and IV.
37. Find the exact value of sin(15°) using angle subtraction.
Answer
sin(15°) = (√6 − √2)/4 — Use sin(45° − 30°) = sin45 cos30 − cos45 sin30.
38. Determine if the identity is true or false: tan x + cot x = sec x csc x.
Answer
True — Rewrite tan x + cot x = (sin/cos + cos/sin) = (sin² + cos²)/sin cos = 1/(sin cos) = sec x csc x.
39. A sinusoidal function has minimum value 2 and maximum value 14. Find its amplitude and midline.
Answer
Amplitude = 6, Midline = 8 — A = (max − min)/2 = (14 − 2)/2 = 6; midline = (max + min)/2 = 8.
40. Solve the equation sin²x = 1/4 for x ∈ [0, 2π].
Answer
x = π/6, 5π/6, 7π/6, 11π/6 — sin x = ±1/2.
41. Prove: 1 + tan²x = sec²x using sine and cosine definitions.
Answer
Proof:
Start with tan x = sin x/cos x ⇒ tan² = sin²/cos².
1 + tan² = (cos² + sin²)/cos² = 1/cos² = sec²x.
42. Evaluate exactly: cos(5π/12).
Answer
cos(5π/12) = (√6 − √2)/4 — Use sum identity cos(π/4 + π/6).
43. Solve for θ: sin(2θ) = 1/2, where 0 ≤ θ ≤ 2π.
Answer
θ = π/12, 5π/12, 13π/12, 17π/12 — 2θ = π/6, 5π/6, 13π/6, 17π/6.
44. The function y = 3 + 4sin((π/3)t − π/6) models a tide. What is its period?
Answer
Period = 6 — Period = 2π / (π/3) = 6.
45. Verify the identity: sin x − sin y = 2 cos((x + y)/2) sin((x − y)/2).
Answer
True — This is the standard sine difference identity.
46. If cos t = 4/5 and t is in Quadrant IV, find tan(2t).
Answer
tan(2t) = −24/7 — Use tan(2t) = 2tan t / (1 − tan²t). Here tan t = −3/4.
47. A sound wave is modeled by S(t) = 12 cos(8πt) − 5. What is its maximum value?
Answer
Max = 7 — 12(1) − 5 = 7.
48. Solve for x: csc x = 2, 0 ≤ x ≤ 2π.
Answer
x = π/6, 5π/6 — sin x = 1/2.
49. Find all values of x where tan x is undefined.
Answer
x = π/2 + kπ, k ∈ ℤ — Cosine equals zero at these points.
50. The temperature over a day is modeled by T(t) = 15 + 8 sin((π/12)t − π/3), where t is in hours since midnight. At what time is the temperature highest?
Answer
t = 10 hours — Maximum occurs when the argument equals π/2: (π/12)t − π/3 = π/2 ⇒ t = 10.