Question

c=85°
a
c=42 ft
b=32°
use the law of sines to solve this triangle. round the lengths of the sides to three decimal places.
a=

°
side a=

ft
side b=

ft

Explanation:

Step1: Find angle A

The sum of angles in a triangle is \(180^\circ\). So, \(A = 180^\circ - B - C\).
Given \(B = 32^\circ\) and \(C = 85^\circ\), we have \(A = 180 - 32 - 85 = 63^\circ\).

Step2: Use Law of Sines to find side a

The Law of Sines states \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
We know \(c = 42\) ft, \(A = 63^\circ\), \(C = 85^\circ\).
So, \(a=\frac{c\sin A}{\sin C}=\frac{42\sin63^\circ}{\sin85^\circ}\).
Calculate \(\sin63^\circ\approx0.8910\), \(\sin85^\circ\approx0.9962\).
Then \(a=\frac{42\times0.8910}{0.9962}\approx\frac{37.422}{0.9962}\approx37.565\) ft.

Step3: Use Law of Sines to find side b

Using \(\frac{b}{\sin B}=\frac{c}{\sin C}\).
So, \(b=\frac{c\sin B}{\sin C}=\frac{42\sin32^\circ}{\sin85^\circ}\).
Calculate \(\sin32^\circ\approx0.5299\).
Then \(b=\frac{42\times0.5299}{0.9962}\approx\frac{22.2558}{0.9962}\approx22.341\) ft.

Answer:

\(A = 63^\circ\)
Side \(a \approx 37.565\) ft
Side \(b \approx 22.341\) ft