Question

Find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say “No triangle.”$a=5,b=3,A=80$

Short answer

The remaining sides and angles are$$\begin{gathered} B=36.2 \\ C=63.8 \\ c=4.55 \end{gathered}$$

Step-by-step solution

Step 1.  Given information     

The given data is$a=5,b=3,A=80$

Step 2. Find the angle B

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

Here, $a=5,b=3,A=80$

$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin 80}{5}=\frac{SinB}{3} \\ SinB=\frac{3*\sin 80}{5} \\ SinB=0.59 \\ B=36.2 \end{gathered}$$

Step 3. Find the angle C

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

Here, $A=80,B=36.2$

$$\begin{gathered} A+B+C=180 \\ 80+36.2+C=180 \\ C=63.8 \end{gathered}$$

Step 4. find the side c 

Use the Law of Sines$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin 80}{5}=\frac{Sin63.8}{c} \\ c=5\frac{\sin 63.8}{\sin 80} \\ c=4.55 \end{gathered}$$