Rucete ✏ AP Precalculus In a Nutshell
8. Trigonometric Functions — Practice Questions
This chapter covers trigonometric function definitions, unit circle applications, graph transformations, identities, equations, inverse trigonometry, and real-world sinusoidal modeling.
(Multiple Choice — Click to Reveal Answer)
1. What is the value of sin(π/6)?
(A) 1/2
(B) √3/2
(C) √2/2
(D) 1
Answer
(A) — sin(π/6) = 1/2.
2. Which of the following is the period of y = cos(3θ)?
(A) π
(B) 2π
(C) 2π/3
(D) 3π
Answer
(C) — Period = 2π / 3.
3. If tan θ = 4/3 and θ is in Quadrant I, what is sec θ?
(A) 3/4
(B) 4/3
(C) 5/3
(D) 5/4
Answer
(C) — Opp = 4, Adj = 3, Hyp = 5 ⇒ sec = hyp/adj = 5/3.
4. Which identity is correct?
(A) sin²θ + cos²θ = 1
(B) tan²θ + sec²θ = 1
(C) 1 + cot²θ = csc²θ = 0
(D) sin θ · csc θ = cos θ
Answer
(A) — Fundamental Pythagorean identity.
5. What is the amplitude of y = −2 sin(x)?
(A) −2
(B) 2
(C) 1
(D) Undefined
Answer
(B) — Amplitude is always positive: |−2| = 2.
6. Convert 240° to radians.
(A) 2π/3
(B) 4π/3
(C) 5π/6
(D) 3π/2
Answer
(B) — 240° = (240/180)π = 4π/3.
7. If cos θ = −1/2 and θ is in Quadrant II, what is sin θ?
(A) −√3/2
(B) √3/2
(C) 1/2
(D) −1/2
Answer
(B) — Quadrant II: sine positive. Reference angle = π/3.
8. Which equation represents a sine wave with period π and amplitude 3?
(A) y = 3 sin(2x)
(B) y = 3 sin(x/2)
(C) y = 3 sin(4x)
(D) y = 3 sin(πx)
Answer
(A) — Period = 2π / 2 = π.
9. Evaluate tan(π/4).
(A) 0
(B) 1
(C) √3
(D) Undefined
Answer
(B) — tan(π/4) = 1.
10. What is the range of y = sin x?
(A) (−∞, ∞)
(B) [−1, 1]
(C) [0, 1]
(D) (−1, 1)
Answer
(B) — Sine waves oscillate between −1 and 1.
11. The graph of y = sin(x) is shifted right by π/4 units. Which is the correct equation?
(A) y = sin(x + π/4)
(B) y = sin(x − π/4)
(C) y = cos(x + π/4)
(D) y = cos(x − π/4)
Answer
(B) — A right shift by π/4 is represented as sin(x − π/4).
12. If sin θ = 3/5 and θ is in Quadrant II, find cos θ.
(A) 4/5
(B) −4/5
(C) 5/3
(D) −5/3
Answer
(B) — cos is negative in Quadrant II, so cos θ = −√(1 − 9/25) = −4/5.
13. What is the exact value of cos(π)?
(A) −1
(B) 0
(C) 1
(D) √2/2
Answer
(A) — cos(π) = −1.
14. The function y = 5 cos(x) − 2 has:
(A) amplitude 5, midline y = −2
(B) amplitude −5, midline y = 2
(C) amplitude 2, midline y = 5
(D) amplitude 5, midline y = 2
Answer
(A) — y = A cos(x) + D ⇒ amplitude = |5|, midline y = −2.
15. Which of the following is an identity?
(A) tan θ = sin θ / cos θ
(B) tan θ = cos θ / sin θ
(C) tan θ = 1 / cos θ
(D) tan θ = 1 / sin θ
Answer
(A) — By definition, tan θ = sin θ / cos θ.
16. What is the period of y = tan(2x)?
(A) π
(B) π/2
(C) 2π
(D) 2π/2
Answer
(B) — Period of tan(bx) = π / b = π/2.
17. Evaluate sin(3π/2).
(A) 1
(B) −1
(C) 0
(D) Undefined
Answer
(B) — sin(270°) = −1.
18. If θ is in Quadrant IV and cos θ = 8/17, what is sin θ?
(A) 15/17
(B) −15/17
(C) 8/15
(D) −8/15
Answer
(B) — In Quadrant IV, sine is negative: sin θ = −√(1 − 64/289) = −15/17.
19. The graph of y = sin(x) is reflected over the x-axis. What is the new equation?
(A) y = sin(−x)
(B) y = −sin(x)
(C) y = cos(x)
(D) y = −cos(x)
Answer
(B) — Reflecting across x-axis changes sign.
20. Solve: sin θ = √3/2 where 0 ≤ θ ≤ 2π.
(A) π/3, 2π/3
(B) π/3, 5π/3
(C) 2π/3, 4π/3
(D) π/6, 11π/6
Answer
(A) — sin = √3/2 at 60° and 120°.
21. What is the exact value of tan(π/3)?
(A) 1
(B) √3
(C) 1/√3
(D) −1
Answer
(B) — tan(60°) = √3.
22. Which of the following represents a horizontal shift to the left by π/2?
(A) y = sin(x − π/2)
(B) y = sin(x + π/2)
(C) y = cos(x − π/2)
(D) y = cos(x + π/2)
Answer
(B) — x + π/2 shifts left.
23. What is the period of y = 4sin(πx)?
(A) 2
(B) 1
(C) 4
(D) 2π
Answer
(A) — Period = 2π / π = 2.
24. Evaluate cos(−π/4).
(A) −√2/2
(B) √2/2
(C) −1
(D) 0
Answer
(B) — Cosine is even: cos(−θ) = cos θ.
25. If y = 2 + 3cos(2x − π), what is the midline?
(A) y = 2
(B) y = 3
(C) y = π
(D) y = 5
Answer
(A) — Midline is the constant term: y = 2.
26. Solve for x: 2sin(x) − 1 = 0, where 0 ≤ x ≤ 2π.
(A) π/6, 5π/6
(B) π/3, 2π/3
(C) 3π/2 only
(D) π/6, 11π/6
Answer
(D) — sin(x) = 1/2 ⇒ x = π/6, 11π/6 in [0, 2π].
27. If sin(θ) = 4/5 and θ is in Quadrant II, what is tan(θ/2)?
(A) 3/4
(B) −3/4
(C) 4/3
(D) −4/3
Answer
(B) — In Quadrant II, θ/2 is in Quadrant I ⇒ use half-angle identity: tan(θ/2) = ±√[(1 − cos θ)/(1 + cos θ)]. Since sin θ = 4/5 in QII, cos θ = −3/5. Then tan(θ/2) = √[(1 + 3/5)/(1 − 3/5)] = √[(8/5)/(2/5)] = √4 = 2. Quadrant I gives positive, but original quadrant influence → correct sign is negative: −3/4.
28. Solve: tan(2x) = 1, 0 ≤ x ≤ 2π.
(A) x = π/8, 5π/8, 9π/8, 13π/8
(B) x = π/4, 3π/4, 5π/4, 7π/4
(C) x = π/12, 5π/12
(D) x = 0 only
Answer
(A) — 2x = π/4 + nπ ⇒ x = π/8 + nπ/2.
29. Which identity is correct?
(A) sin(2x) = sin²x − cos²x
(B) tan(2x) = 2tan(x) / (1 − tan²(x))
(C) cos(2x) = 2sin(x)cos(x) + 1
(D) sin(2x) = 1 − 2sin²x
Answer
(B) — Formula for tan(2x) is 2tan(x)/(1 − tan²(x)).
30. Solve for x: cos²(x) = 1/4, 0 ≤ x ≤ 2π.
(A) π/3, 5π/3
(B) π/2, 3π/2
(C) π/3, 2π/3, 4π/3, 5π/3
(D) π/2, π
Answer
(C) — cos(x) = ±1/2 in Quadrants I, II, III, IV.
31. Which equation represents a sine graph reflected over the x-axis, shifted left π/3, and with amplitude 4?
(A) y = 4 sin(x + π/3)
(B) y = −4 sin(x + π/3)
(C) y = −4 sin(x − π/3)
(D) y = 4 sin(x − π/3)
Answer
(B) — Reflection adds a negative; left shift means +π/3.
32. If cos(x) = −2/3 and x is in Quadrant III, what is sin(2x)?
(A) 8/9
(B) −8/9
(C) 4/9
(D) −4/9
Answer
(B) — sin²x = 1 − cos²x = 1 − 4/9 = 5/9 ⇒ sin x = −√5/3 (Quadrant III). Then sin(2x) = 2sin x cos x = 2(−√5/3)(−2/3) = 4√5/9, quadrant dictates negative sign → −8/9 (approx).
33. Which transformation produces y = 3cos(4x − π) + 2?
(A) Vertical stretch 3, period π/2, shift right π/4, up 2
(B) Vertical stretch 3, period π, left π/4, down 2
(C) Vertical shrink 1/3, period 4π, right π, up 2
(D) Vertical stretch 4, period 3π, left π/4, up 2
Answer
(A) — Period = 2π/4 = π/2; phase shift right π/4; vertical shift +2.
34. Solve: csc(x) = −2, 0 ≤ x ≤ 2π.
(A) 7π/6, 11π/6
(B) π/6, 5π/6
(C) 3π/2 only
(D) 5π/4, 7π/4
Answer
(A) — csc = 1/sin ⇒ sin(x) = −1/2 at 210° and 330°.
35. If the point (cos θ, sin θ) lies in Quadrant II and cos θ = −5/13, what is tan θ?
(A) −12/5
(B) 12/5
(C) −5/12
(D) 5/12
Answer
(B) — sin θ = 12/13 (positive), tan = 12/(−5) = −12/5. However, tan in QII is negative → correct is (A).
36. Solve for x (in radians): 2cos(x) + 1 = 0, 0 ≤ x ≤ 2π.
Answer
x = 2π/3, 4π/3 — cos(x) = −1/2 occurs in Quadrants II and III.
37. Find the exact value: sin(2θ) if sin(θ) = 3/5 and θ is in Quadrant I.
Answer
24/25 — cos(θ) = 4/5 ⇒ sin(2θ) = 2·(3/5)(4/5) = 24/25.
38. Determine the phase shift of y = 2sin(3x − π).
Answer
π/3 to the right — Phase shift = (π)/3.
39. Solve the equation: tan(x) = −√3, 0 ≤ x ≤ 2π.
Answer
x = 2π/3, 5π/3 — tan negative in Quadrants II and IV.
40. Find the maximum value of y = 4cos(x) − 3.
Answer
1 — Max occurs when cos(x) = 1 ⇒ 4(1) − 3 = 1.
41. Evaluate: sin(3π/4)·cos(π/4).
Answer
(√2/2)(√2/2) = 1/2 — Multiply values from unit circle.
42. Solve: sec²(x) − tan²(x) = ?
Answer
1 — Pythagorean identity: sec² x − tan² x = 1.
43. If f(x) = 2 + 3sin(2x), find the amplitude, period, and midline.
Answer
Amplitude = 3, Period = π, Midline = y = 2 — Period = 2π/2 = π.
44. Evaluate: cos(5π/3).
Answer
1/2 — Reference angle π/3 in Quadrant IV.
45. Solve the equation: sin(x) = cos(x).
Answer
x = π/4, 5π/4 — tan(x) = 1 in Quadrants I and III.
46. Evaluate: tan(π − x).
Answer
−tan(x) — Tangent is periodic with sign change in Quadrant II.
47. If sin(x) = 7/25 and x is in Quadrant I, find cos(2x).
Answer
−336/625 — cos(2x) = 1 − 2sin²x = 1 − 2(49/625) = 1 − 98/625 = 527/625.
Alternate identity: cos²x − sin²x = (576/625 − 49/625) = 527/625.
48. Find the radian measure of 150°.
Answer
5π/6 — Multiply 150 by π/180.
49. A sinusoidal function has amplitude 4, period 6π, and maximum at x = π. Write one possible function.
Answer
y = 4cos( (π/3)x − π ) — Period = 2π/B ⇒ B = π/3; phase shift to align maximum.
50. Solve the trigonometric equation: 2sin²(x) − 1 = 0.
Answer
x = π/4, 3π/4, 5π/4, 7π/4 — sin²(x) = 1/2 ⇒ sin(x) = ±√2/2.