Question

Find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say “No triangle.”$a=4,A=20,B=100$

Short answer

The remaining sides and angles are$$\begin{gathered} C=60 \\ b=11.5 \\ c=10.1 \end{gathered}$$$$\begin{gathered} C=60 \\ b=11.5 \\ c=10.1 \end{gathered}$$

Step-by-step solution

Step 1, Given information   

The given data is$a=4,A=20,B=100$

Step 2. Find the angle C

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

$$\begin{gathered} A+B+C=180 \\ 20+100+C=180 \\ C=180-120 \\ C=60 \end{gathered}$$

Step 3. Find the length of side b

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

Here, $a=4,A=20,B=100,C=60$

$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin 20}{4}=\frac{Sin100}{b} \\ b=\frac{4Sin100}{\sin 20} \\ b=11.5 \end{gathered}$$

Step 4. Find the length of sides c

Use the Law of Sines ,$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin 20}{4}=\frac{Sin60}{c} \\ c=\frac{4*Sin60}{\sin 20} \\ c=10.1 \end{gathered}$$