Right Triangle Trigonometry Test Answer Key with Scoring – Geometry

Question 1 Three Ratios (3 points)

(a) sin A = 3537.

(b) cos A = 1237.

(c) For angle B, opposite is AC = 12, adjacent is BC = 35: tan B = 1235.

Scoring: 1 point per part.

Question 2 Complementary Equation (2 points)

Sine equals cosine of the complement, so (2x + 4) + (3x − 4) = 90. Then 5x = 90 and x = 18.

Check: sin 40° = cos 50°, correct.

Scoring: 1 point for the complementary relationship, 1 point for the answer.

Question 3 Find a Side (2 points)

BC is opposite the 58° angle and AB is the hypotenuse: sin 58° = 18.4x.

x = 18.4sin 58° ≈ 18.40.8480 ≈ 21.7.

x ≈ 21.7

Scoring: 1 point for the correct ratio, 1 point for the answer.

Question 4 Find an Angle (2 points)

The horizontal distance is adjacent to the angle and the ramp is the hypotenuse: cos θ = 1725 = 0.68.

θ = cos−1(0.68) ≈ 47.16°.

θ ≈ 47.2°

Scoring: 1 point for the ratio, 1 point for the answer.

Question 5 Special Right Triangle (2 points)

Short leg = 26 ÷ 2 = 13. Long leg = 13√3.

13 in and 13√3 in

Scoring: 1 point for each leg.

Question 6 Solve the Triangle (3 points)

B = 90° − 41° = 49°.

BC = 18 · tan 41° ≈ 18 · 0.8693 = 15.65. AB = 18cos 41° ≈ 180.7547 ≈ 23.85.

B = 49°, BC ≈ 15.6, AB ≈ 23.9

Scoring: 1 point for B, 1 point for BC, 1 point for AB.

Question 7 Height of a Building (3 points)

Height above eye level = 65 · tan 47° ≈ 65 · 1.0724 = 69.7 ft. Add 6 ft: 75.7 ft.

≈ 76 ft

Scoring: 1 point for the ratio, 1 point for the product, 1 point for adding the eye height.

Question 8 Two Buoys (3 points)

Horizontal distances: 150tan 21° ≈ 390.8 ft and 150tan 36° ≈ 206.5 ft.

Difference: 390.8 − 206.5 ≈ 184.3.

≈ 184 ft

Scoring: 1 point for each distance, 1 point for the difference.