Question

Find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say “No triangle.”$c=5,b=4,A=70$

Short answer

The remaining sides and angles are$$\begin{gathered} a=5.22 \\ B=46 \\ C=64 \end{gathered}$$

Step-by-step solution

Step 1.  Given information    

The given data is$c=5,b=4,A=70$

Step 2. Find the length of side a

Use the cosines rule of Law,$\cos A=\frac{b^2+c^2-a^2}{2bc}$

Here, $c=5,b=4,A=70$

$$\begin{gathered} \cos A=\frac{b^2+c^2-a^2}{2bc} \\ \cos 70=\frac{4^2+5^2-a^2}{2*4*5} \\ 0.34=\frac{16+25-a^2}{40} \\ 0.34*40=41-a^2 \\ {a}^2=41-13.68 \\ {a}^2=27.32 \\ a=5.22 \end{gathered}$$

Step 3.Find the angle B

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

Here,$A=70,a=5.22,b=4,c=5$

$$\begin{gathered} \frac{\sin A}{a}=\frac{SinB}{b}=\frac{SinC}{c} \\ \frac{\sin 70}{5.22}=\frac{SinB}{4} \\ SinB=4\frac{\sin 70}{5.22} \\ SinB=0.72 \\ B=46 \end{gathered}$$

Step 4. Find the angle C

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

$$\begin{gathered} A+B+C=180 \\ 70+46+C=180 \\ C=180-116 \\ C=64 \end{gathered}$$