Inverse Trigonometric Functions Test Answer Key with Scoring – Precalculus

Question 1 Exact arcsin and arccos (2 points)

(a) The angle in [−π/2, π/2] with sine −√2/2 is −π/4.

(b) The angle in [0, π] with cosine −1/2 is 2π/3.

Scoring: 1 point per value.

Question 2 arctan and Ranges (2 points)

(a) The angle in (−π/2, π/2) with tangent √3 is π/3.

(b) Domain all real numbers; range (−π/2, π/2).

Scoring: 1 point per part.

Question 3 Compositions (2 points)

(a) Opposite 8, hypotenuse 17, adjacent 15: 15/17.

(b) sin(4π/3) = −√3/2 and arcsin(−√3/2) = −π/3.

Scoring: 1 point per part.

Question 4 Read Solutions from a Graph (2 points)

1. The line meets the curve two times.
2. Cosine is −1/2 with reference angle π/3 in Quadrants II and III: x = 2π/3 and 4π/3.

Scoring: 1 point per part.

Question 5 A Quadratic in Sine (3 points)

Factor: (2 sin x + 1)(sin x + 1) = 0.

sin x = −1/2 gives x = 7π/6 and 11π/6. sin x = −1 gives x = 3π/2.

x = 7π/6, 3π/2, 11π/6.

Scoring: 1 point for the factorization; 1 point for each correct branch.

Question 6 Tangent of a Double Angle (3 points)

tan u = √3 when u = π/3 + kπ, so 2x = π/3 + kπ and x = π/6 + kπ/2.

On [0, 2π): x = π/6, 2π/3, 7π/6, 5π/3. General: x = π/6 + kπ/2.

Scoring: 1 point for the general form of the angle; 1 point for the four solutions; 1 point for the general solution.

Question 7 Calculator Solution (2 points)

cos x = 0.45. arccos 0.45 ≈ 1.1040. The second solution is 2π − 1.1040 ≈ 5.1792.

Scoring: 1 point for each solution.

Question 8 Wave Buoy Height (3 points)

1. 6 + 2.4cos(πt/5) = 8, so cos(πt/5) = 2/2.4 ≈ 0.8333.
2. πt/5 = arccos(0.8333) ≈ 0.5857, so t ≈ 0.93 s.
3. By symmetry t = 10 − 0.93 ≈ 9.07 s.

Scoring: 1 point per part.

Question 9 A False Identity (1 point)

sin π = 0, and arcsin 0 = 0, not π. The output of arcsin must lie in [−π/2, π/2], and π is outside that interval.

Scoring: 1 point for a correct explanation with the numbers.