Radians and Unit Circle Test Answer Key with Scoring – Algebra 2

Question 1 Convert angle measures (2 points)

1. 210π/180 = 7π/6.
2. 5 · 180/9 = 100°.

Scoring: 1 point each.

Question 2 Exact values (3 points)

1. Reference angle π/3, quadrant II: √3/2.
2. Reference angle π/4, quadrant III: −√2/2.
3. Point (√3/2, −1/2), so tan = (−1/2) ÷ (√3/2) = −√3/3.

Scoring: 1 point each.

Question 3 Read the unit circle (3 points)

1. The reference angle is π/3 and the point is in quadrant II, so θ = 2π/3.
2. cos = −1/2, sin = √3/2, tan = (√3/2) ÷ (−1/2) = −√3.

Scoring: 1 point for the angle, 1 point for sin and cos, 1 point for tan.

Question 4 Arc length (2 points)

s = 14 · 11π/18 = 77π/9 ≈ 26.88. 77π/9 ft ≈ 26.88 ft.

Scoring: 1 point for the exact value, 1 point for the decimal.

Question 5 Ratios from a given sine (2 points)

cos² θ = 1 − 81/1681 = 1600/1681, so cos θ = −40/41 in quadrant III. Then tan θ = (−9/41) ÷ (−40/41). cos θ = −40/41 and tan θ = 9/40.

Scoring: 1 point for cos, 1 point for tan.

Question 6 Ratios from a given cosine (2 points)

sin² θ = 1 − 7/16 = 9/16, so sin θ = 3/4. Then tan θ = (3/4) ÷ (−√7/4) = −3/√7 = −3√7/7. sin θ = 3/4 and tan θ = −3√7/7.

Scoring: 1 point for sin, 1 point for tan.

Question 7 Coterminal angles (2 points)

19π/6 − 2π = 7π/6. For 7π/6 the reference angle is π/6 and the quadrant is III, so cosine is negative. cos(19π/6) = −√3/2.

Scoring: 1 point for the coterminal angle, 1 point for the value.

Question 8 Verify an identity (2 points)

Multiply: 1 − cos² θ. From sin² θ + cos² θ = 1, we get 1 − cos² θ = sin² θ. Shown.

Scoring: 1 point for the product, 1 point for the identity.

Question 9 Ferris wheel (2 points)

Height = 40 sin(2π/3) = 40 · √3/2 = 20√3 ≈ 34.64. 20√3 ft ≈ 34.6 ft.

Scoring: 1 point for the exact value, 1 point for the decimal.