Unit Circle Test Answer Key with Scoring – Precalculus

Question 1 Convert Angle Measures (2 points)

(a) 315π/180 = 7π/4.

(b) 7π/18 · 180/π = 70°.

Scoring: 1 point per conversion.

Question 2 Sprinkler Arc (2 points)

1. s = 18 · 5π/6 = 15π ≈ 47.1 feet.
2. θ = 30/18 = 5/3 rad = 5/3 · 180/π ≈ 95.5°.

Scoring: 1 point per part.

Question 3 Point on the Unit Circle (2 points)

1. The reference angle is π/4 and both coordinates are negative: (−√2/2, −√2/2).
2. tan θ = sin ÷ cos = 1.

Scoring: 1 point per part.

Question 4 Exact Trigonometric Values (3 points)

(a) Quadrant II, reference π/4: −√2/2.

(b) sin(−7π/6) = −sin(7π/6) = −(−1/2) = 1/2.

(c) Quadrant IV, reference π/3: (−√3/2) ÷ (1/2) = −√3.

Scoring: 1 point per value.

Question 5 Find Cosine and Tangent (2 points)

cos² θ = 1 − 25/169 = 144/169. In Quadrant III cosine is negative, so cos θ = −12/13.

tan θ = (−5/13) ÷ (−12/13) = 5/12.

Scoring: 1 point for cos θ with the correct sign; 1 point for tan θ.

Question 6 Reference Angle and Sine (2 points)

17π/6 − 2π = 5π/6, which is in Quadrant II.

(a) Reference angle: π − 5π/6 = π/6.

(b) Sine is positive in Quadrant II: 1/2.

Scoring: 1 point per part.

Question 7 Tire Rotation (3 points)

1. ω = 420 · 2π = 840π ≈ 2,638.9 rad/min.
2. v = 15 · 840π = 12,600π ≈ 39,584 in/min.
3. 39,584 × 60 ÷ 63,360 ≈ 37.5 mph.

Scoring: 1 point per part.

Question 8 Symmetry of Marked Points (2 points)

1. Cosine is even and sine is odd: B = (cos 0.8, −sin 0.8) ≈ (0.697, −0.717).
2. cos(π − 0.8) = −cos 0.8 and sin(π − 0.8) = sin 0.8, so C ≈ (−0.697, 0.717).

Scoring: 1 point per point, both coordinates required.

Question 9 Particle Motion (2 points)

(a) θ = 7π/3 = 2π + π/3, so the position is (1/2, √3/2).

(b) The point (−1, 0) is at angle π: (π/3)t = π gives t = 3 seconds.

Scoring: 1 point per part.