Question

1. determine the possible measurement(s) of angle b, given that in \\( \triangle a b c, \angle a=32^{\circ}, a = 8.5 \mathrm{~cm} \\) and \\( b = 12 \mathrm{~cm} \\). round your answer(s) to the nearest degree. 4

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}\).
Substituting the given values: \(\frac{\sin 32^{\circ}}{8.5}=\frac{\sin B}{12}\).

Step2: Solve for \(\sin B\)

Cross - multiply to get \(\sin B=\frac{12\times\sin 32^{\circ}}{8.5}\).
Calculate \(\sin 32^{\circ}\approx0.5299\).
Then \(\sin B=\frac{12\times0.5299}{8.5}\approx\frac{6.3588}{8.5}\approx0.7481\).

Step3: Find the principal value of \(B\)

Using the inverse sine function, \(B = \sin^{-1}(0.7481)\approx48^{\circ}\).

Step4: Consider the supplementary angle

Since \(\sin\theta=\sin(180^{\circ}-\theta)\), another possible value of \(B\) is \(180^{\circ}- 48^{\circ}=132^{\circ}\).
Check the triangle - angle sum:
For \(B = 48^{\circ}\), \(C=180^{\circ}-(32^{\circ}+48^{\circ}) = 100^{\circ}\).
For \(B = 132^{\circ}\), \(C=180^{\circ}-(32^{\circ}+132^{\circ})=16^{\circ}\). Both are valid as angles in a triangle.

Answer:

\(B = 48^{\circ}\) or \(B = 132^{\circ}\)