Geometry Law of Sines and Law of Cosines Test Solutions and Scoring (PDF)

\(48 = \frac{1}{2} \cdot 12 \cdot 10 \cdot \text{sin}\,C\)

\(\text{sin}\,C = \frac{48}{60} = 0.8\)

\(C = \text{sin}^{-1}(0.8) = 53.1^{\circ}\)

C = 53.1°

Scoring: 1 point for the formula, 1 point for the angle.

\(\frac{b}{\text{sin}\,58^{\circ}} = \frac{25}{\text{sin}\,49^{\circ}}\)

\(b = 28.1\)

b = 28.1

Scoring: 1 point for the setup, 1 point for b.

1. \(c^{2} = 8^{2} + 11^{2} – 2 \cdot 8 \cdot 11 \cdot \text{cos}\,75^{\circ}\)
2. \(c^{2} = 185 – 176 \cdot 0.2588 = 139.4\), so \(c = 11.8\)

Scoring: (a) 1 point. (b) 1 point for the substitution, 1 point for c.

\(7^{2} = 6^{2} + 10^{2} – 2 \cdot 6 \cdot 10 \cdot \text{cos}\,B\)

\(\text{cos}\,B = \frac{36 + 100 – 49}{120} = 0.725\)

\(B = 43.5^{\circ}\)

43.5°

Scoring: 1 point for the setup, 1 point for the angle.

1. The Law of Cosines.
2. The Law of Sines (after finding the third angle if needed).

Scoring: 1 point per part.

\(16^{2} = 9^{2} + 12^{2} – 2 \cdot 9 \cdot 12 \cdot \text{cos}\,C\)

\(\text{cos}\,C = \frac{81 + 144 – 256}{216} = -0.1435\)

The cosine is negative, so the largest angle is obtuse (about 98.3°).

Obtuse

Scoring: 1 point for the setup, 1 point for the conclusion.

\(d^{2} = 15^{2} + 22^{2} – 2 \cdot 15 \cdot 22 \cdot \text{cos}\,36^{\circ}\)

\(d^{2} = 709 – 660 \cdot 0.8090 = 175.0\)

\(d = 13.2\)

13.2 meters

Scoring: 1 point for the setup, 1 point for the substitution, 1 point for d.

\(\frac{1}{2} \cdot 14 \cdot 20 \cdot \text{sin}\,70^{\circ} = 140 \cdot 0.9397 = 131.6\)

131.6 square blocks

Scoring: 1 point for the formula, 1 point for the area.

The 67° angle is largest, and 20 is the longest side, so the pattern holds.

The smallest angle, 52°, faces the shortest side, 17.1.

Scoring: 1 point each.