Question

solve for the remaining angles and side of the two triangles that can be created. round to the nearest hundredth: a = 60°, a = 7, b = 8

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Substitute \(A = 60^{\circ}\), \(a = 7\), and \(b = 8\) into the formula: \(\frac{7}{\sin60^{\circ}}=\frac{8}{\sin B}\).
Cross - multiply to get \(7\sin B=8\sin60^{\circ}\).
Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\), then \(7\sin B = 8\times0.866 = 6.928\).
So, \(\sin B=\frac{6.928}{7}\approx0.9897\).

Step2: Find angle \(B\)

Since \(\sin B\approx0.9897\), then \(B=\sin^{- 1}(0.9897)\approx81.11^{\circ}\) or \(B = 180^{\circ}-81.11^{\circ}=98.89^{\circ}\) (because \(\sin\theta=\sin(180^{\circ}-\theta)\)).

Step3: Find angle \(C\) for the first case (\(B = 81.11^{\circ}\))

Using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), when \(A = 60^{\circ}\) and \(B = 81.11^{\circ}\), then \(C=180^{\circ}-(60^{\circ}+81.11^{\circ})=38.89^{\circ}\).

Step4: Find side \(c\) for the first case (\(B = 81.11^{\circ}\))

Using the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
Substitute \(a = 7\), \(A = 60^{\circ}\), and \(C = 38.89^{\circ}\) (where \(\sin38.89^{\circ}\approx0.628\) and \(\sin60^{\circ}\approx0.866\)).
\(c=\frac{7\times\sin38.89^{\circ}}{\sin60^{\circ}}=\frac{7\times0.628}{0.866}\approx5.07\).

Step5: Find angle \(C\) for the second case (\(B = 98.89^{\circ}\))

Using the angle - sum property of a triangle (\(A + B + C = 180^{\circ}\)), when \(A=60^{\circ}\) and \(B = 98.89^{\circ}\), then \(C=180^{\circ}-(60^{\circ}+98.89^{\circ})=21.11^{\circ}\).

Step6: Find side \(c\) for the second case (\(B = 98.89^{\circ}\))

Using the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
Substitute \(a = 7\), \(A = 60^{\circ}\), and \(C = 21.11^{\circ}\) (where \(\sin21.11^{\circ}\approx0.360\) and \(\sin60^{\circ}\approx0.866\)).
\(c=\frac{7\times\sin21.11^{\circ}}{\sin60^{\circ}}=\frac{7\times0.360}{0.866}\approx2.91\).

Answer:

First triangle:

• \(B\approx81.11^{\circ}\), \(C\approx38.89^{\circ}\), \(c\approx5.07\)

Second triangle:

• \(B\approx98.89^{\circ}\), \(C\approx21.11^{\circ}\), \(c\approx2.91\)