Question

1. find the measure of the indicated angle, to the nearest degree.5 solution

Explanation:

Step1: Use the Law of Cosines

First, find side $AC$ using the Law of Cosines: $c^{2}=a^{2}+b^{2}-2ab\cos C$. Here, $a = 13$, $b = 16$, and $C=65^{\circ}$.
\[

$$\begin{align*} c^{2}&=13^{2}+16^{2}-2\times13\times16\times\cos65^{\circ}\\ c^{2}&=169 + 256-416\times0.4226\\ c^{2}&=169+256 - 176.79\\ c^{2}&=248.21\\ c&\approx15.76 \end{align*}$$

\]

Step2: Use the Law of Sines

Then, use the Law of Sines $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$ to find $\angle A$. We know $a = 13$, $c\approx15.76$, and $C = 65^{\circ}$.
\[

$$\begin{align*} \frac{\sin A}{13}&=\frac{\sin65^{\circ}}{15.76}\\ \sin A&=\frac{13\times\sin65^{\circ}}{15.76}\\ \sin A&=\frac{13\times0.9063}{15.76}\\ \sin A&=\frac{11.7819}{15.76}\\ \sin A&\approx0.7476 \end{align*}$$

\]

Step3: Find the angle

$A=\sin^{- 1}(0.7476)\approx48^{\circ}$

Answer:

$48^{\circ}$