Question

use the law of sines to solve for the indicated values. round decimal answers to the nearest tenth.
a) ( mangle b=) type your answer... ( ^{circ})
b) ( a=) type your answer... in
c) ( b=) type your answer... in

Explanation:

Step1: Find \(m\angle B\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(m\angle B=180^{\circ}-(83^{\circ}+36^{\circ})\).

$$m\angle B = 180^{\circ}-119^{\circ}=61^{\circ}$$

Step2: Use the Law of Sines to find \(a\)

The Law of Sines states \(\frac{a}{\sin A}=\frac{c}{\sin C}\). Here, \(c = 27\) in, \(A = 36^{\circ}\), \(C=83^{\circ}\).

$$a=\frac{27\times\sin36^{\circ}}{\sin83^{\circ}}$$

$$a=\frac{27\times0.5878}{0.9925}$$

$$a=\frac{15.8706}{0.9925}\approx16.0$$

Step3: Use the Law of Sines to find \(b\)

Using the Law of Sines \(\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, \(B = 61^{\circ}\), \(c = 27\) in, \(C=83^{\circ}\)

$$b=\frac{27\times\sin61^{\circ}}{\sin83^{\circ}}$$

$$b=\frac{27\times0.8746}{0.9925}$$

$$b=\frac{23.6142}{0.9925}\approx23.8$$

Answer:

a) \(61\)
b) \(16.0\)
c) \(23.8\)