Question

Find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say “No triangle.” $a=3,c=1,B=100$

Short answer

The value of$$\begin{gathered} A=62.8 \\ b=3.32 \\ C=17.2 \end{gathered}$$

Step-by-step solution

Step 1.  Given information  

The given data is$a=3,c=1,B=100$

Step 2. Find the side  b

Use cosines rule $\cos B=\frac{a^2+c^2-b^2}{2ac}$

Here, $a=3,c=1,B=100$

$$\begin{gathered} \cos B=\frac{a^2+c^2-b^2}{2ac} \\ \cos 100=\frac{3^2+1^2-b^2}{2*3*1} \\ \left(-0.17\right)*6=10-b^2 \\ {b}^2=10+1.02 \\ {b}^2=11.02 \\ b=3.32 \end{gathered}$$

Step 3. Find the angle A

Using The Law of Sines $\frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c}$

$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin A}{3}=\frac{Sin100}{3.32} \\ \sin A=0.89 \\ A=62.8 \end{gathered}$$

Step 4.Find the angle C

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

$$\begin{gathered} A+B+C=180 \\ 62.8+100+C=180 \\ C=180-162.8 \\ C=17.2 \end{gathered}$$