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

\(\text{hyp}=\frac{15}{\cos28^{\circ}}\approx 16.99\). \(\text{opp}=15\tan28^{\circ}\approx 7.98\).

\(16.99\) and \(7.98\)

Scoring: 1 point each.

\(b=\frac{12\sin75^{\circ}}{\sin45^{\circ}}\approx 16.39\).

\(b\approx 16.39\)

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

(a) \(\sin B=\frac{10\sin30^{\circ}}{4}=1.25>1\): none. (b) \(\sin B=\frac{10\sin40^{\circ}}{8}\approx 0.8035\): \(B\approx 53.5^{\circ}\) or \(126.5^{\circ}\).

(a) none; (b) \(53.5^{\circ}\) or \(126.5^{\circ}\)

Scoring: 1 point for (a), 1 point for (b).

\(c^{2}=36+81-2(6)(9)\cos60^{\circ}=117-54=63\), so \(c=\sqrt{63}\approx 7.94\).

\(c\approx 7.94\)

Scoring: 1 point for c squared, 1 point for c.

\(\cos A=\frac{100+169-64}{260}\), \(A\approx 37.96^{\circ}\). \(\cos B=\frac{64+169-100}{208}\), \(B\approx 50.25^{\circ}\). \(C=180^{\circ}-A-B\approx 91.79^{\circ}\).

\(37.96^{\circ}\), \(50.25^{\circ}\), \(91.79^{\circ}\)

Scoring: 1 point for each angle.

\(\frac12(7)(10)\sin50^{\circ}\approx 26.81\).

\(26.81\) square units

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

\(s=18\). Area \(=\sqrt{18\cdot9\cdot8\cdot1}=\sqrt{1296}=36\).

36 square units

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

(a) \(d^{2}=16+36-48\cos50^{\circ}\approx 21.15\), \(d\approx 4.60\). (b) \(\frac12(4)(6)\sin50^{\circ}\approx 9.19\). (c) \(\cos M=\frac{16+d^{2}-36}{2\cdot4\cdot d}\), so \(M\approx 88.2^{\circ}\).

\(4.60\text{ mi}\); \(9.19\text{ mi}^{2}\); \(88.2^{\circ}\)

Scoring: 1 point for each part.

\(\cos C=\frac{49+81-144}{126}=-\frac{1}{9}\), so \(C\approx 96.38^{\circ}\).

\(96.38^{\circ}\)

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