Question

Find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say “No triangle.”$a=1,b=3,C=40$

Short answer

The remaining angles and sides of a triangle is$$\begin{gathered} c=2.32 \\ A=16 \\ B=124 \end{gathered}$$

Step-by-step solution

Step 1.  Given information   

The given data is$a=1,b=3,C=40$

Step 2. find the side c

Use cosines Rule $\cos C=\frac{a^2+b^2-c^2}{2ab}$

Here. $a=1,b=3,C=40$

$$\begin{gathered} \cos C=\frac{a^2+b^2-c^2}{2ab} \\ \cos 40=\frac{1^2+3^2-c^2}{2*1*3} \\ \cos 40=\frac{10-c^2}{6} \\ 10-c^2=4.6 \\ {c}^2=5.4 \\ c=2.32 \end{gathered}$$

Step 3. Find the angle B

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

Here, $a=1,b=3,c=2.32,C=40$

$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c}. \\ \frac{\sin A}{1}=\frac{Sin40}{2.32} \\ \sin A=0.27 \\ A=16 \end{gathered}$$

Step 4. Find the angle B

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

$$\begin{gathered} A+B+C=180 \\ 16+B+40=180 \\ B=180-56 \\ B=124 \end{gathered}$$