Question

In the given problem solve the triangle using either the law of sines or law of cosines-

$A=65°,B=72°,b=7$

Short answer

Required values of the triangle are

$C=43°,a\approx 6.58,c\approx 4.97$

Step-by-step solution

Step 1. Given information

In the given triangle,

$A=65°,B=72°,b=7$

Law of sines is given by,

$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Step 2. Calculation

We know, sum of all angles of a triangle i.e.

$$\begin{gathered} A+B+C=180° \\ \Rightarrow C=180°-65°-72° \\ \Rightarrow C=43° \end{gathered}$$

For $a$,

$$\begin{gathered} \frac{\sin A}{a}=\frac{\sin B}{b} \\ \Rightarrow \frac{\sin 65°}{a}=\frac{\sin 72°}{7} \\ \Rightarrow \frac{0.9}{0.95}\times 7=a \\ \Rightarrow a\approx 6.58 \end{gathered}$$

For $c$,

$$\begin{gathered} \frac{\sin B}{b}=\frac{\sin C}{c} \\ \Rightarrow \frac{\sin 72°}{7}=\frac{\sin 43°}{c} \\ \Rightarrow c=\frac{0.68}{0.95}\times 7 \\ \Rightarrow c\approx 4.97 \end{gathered}$$