Question

Find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say “No triangle.”$a=3,A=10,b=4$

Short answer

The remaining sides and angles are$$\begin{gathered} B=13.3 \\ C=156.6 \\ c=6.86 \end{gathered}$$

Step-by-step solution

Step 1. Given information   

The given data is$a=3,A=10,b=4$

Step 2. Find the angle B

Use the Law of sines, $\frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c}$

Here, $a=3,A=10,b=4$

$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin 10}{3}=\frac{SinB}{4} \\ SinB=\frac{4*\sin 10}{3} \\ SinB=0.23 \\ B=13.3 \end{gathered}$$

Step 3. Find the angle C 

Since the angle A, B, and c are the angel of a triangle,

$$\begin{gathered} A+B+C=180 \\ 10+13.3+C=180 \\ C=156.6 \end{gathered}$$

Step 4. Find the length of  side C 

Use the Law of sines, $\frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c}$

Here, $a=3,A=10,C=156.6$

$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin 10}{3}=\frac{Sin156.6}{c} \\ c=\frac{3Sin156.6}{\sin 10} \\ c=6.8 \end{gathered}$$