Question

Solve each triangle.

Short answer

The value of$$\begin{gathered} A=66.5 \\ B=23.5 \\ a=4.58 \end{gathered}$$

Step-by-step solution

Step 1. Given information 

The given data from the graph is$$\begin{gathered} b=2 \\ c=5 \end{gathered}$$

Step 2. Find angle B

Applying the law of sins we get $$\begin{gathered} \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \\ \frac{\sin B}{2}=\frac{\sin 90}{5} \\ \sin B=\frac{2}{5}=0.4 \\ B=23.5^{\circ } \end{gathered}$$

Step 3. Find the angle A

we know sums of all the angles is 180

Then$$\begin{gathered} A+90+23.5=180 \\ A=180-113.5 \\ A=66.5 \end{gathered}$$

Step 4. Find the length of side a

By applying law of sines we get$$\begin{gathered} \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \\ \frac{\sin 66.5}{a}=\frac{\sin 23.5}{2}=\frac{\sin 90}{5} \\ \frac{\sin 66.5}{a}=\frac{\sin 90}{5} \\ \frac{\sin 66.5}{a}=\frac{1}{5} \\ a=5\sin 66.5 \\ a=4.58 \end{gathered}$$