Question

christopher is solving the triangle by finding all the missing sides and angles.
part 1:
$x^{2}=8^{2}+14^{2}-2(8)(14)\cos 67$
$x^{2}=64 + 196 - 224\cos 67$
$x^{2}=172.5$
$x = 13.1$
part 2: (angle opposite 8)
$\frac{\sin 67}{13.1}=\frac{\sin \theta}{8}$
$\sin 67\cdot 8 = 13.1\sin \theta$
$12.9 = 13.1\sin \theta$
$0.9837 = \sin \theta$
$\theta = 76.6$
part 3: (angle opposite 14)
$\frac{\sin 67}{13.1}=\frac{\sin \theta}{14}$
$\sin 67\cdot 13.1 = 14\sin \theta$
$12.1 = 14\sin \theta$
$0.8613 = \sin \theta$
$\theta = 59.5$

Explanation:

Step1: Law of Cosines

The Law of Cosines \(c^{2}=a^{2}+b^{2}-2ab\cos C\) is used to find side \(x\). Here \(a = 8\), \(b=14\), \(C = 67^{\circ}\).

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Step2: Law of Sines for angle opposite \(8\)

The Law of Sines \(\frac{\sin A}{a}=\frac{\sin B}{b}\) is used. Let \(A = 67^{\circ}\), \(a = 13.1\), \(b = 8\).

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Step3: Law of Sines for angle opposite \(14\)

Using the Law of Sines again. Let \(A = 67^{\circ}\), \(a = 13.1\), \(b = 14\).

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Answer:

In Part 2, the error is using \(12.9 = 13.1\sin\theta\) (should be \(8\times\sin67^{\circ}\approx7.36\) instead of \(12.9\)). The correct angle opposite \(8\) is approximately \(34.2^{\circ}\). In Part 3, the error is using \(\sin67\cdot13.1\) (should be \(\sin67\cdot14\) in the cross - multiplication). The correct angle opposite \(14\) is approximately \(79.6^{\circ}\).