Question

find the length of side a.
c = 20° a = ? c =
round to the nearest tenth.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. In this triangle, we know angle $A = 120^{\circ}$, angle $C=20^{\circ}$, and side $b = 8$ (assuming side opposite angle $B$ is 8, since angle $B = 40^{\circ}$ as $180-(120 + 20)=40^{\circ}$). We want to find side $a$. Using $\frac{a}{\sin A}=\frac{b}{\sin B}$, substitute $A = 120^{\circ}$, $B = 40^{\circ}$, and $b = 8$. So, $a=\frac{8\times\sin120^{\circ}}{\sin40^{\circ}}$.

Step2: Calculate the sine values

We know that $\sin120^{\circ}=\frac{\sqrt{3}}{2}\approx0.866$ and $\sin40^{\circ}\approx0.643$. Then $a=\frac{8\times0.866}{0.643}$.

Step3: Perform the division

$8\times0.866 = 6.928$. Then $a=\frac{6.928}{0.643}\approx10.8$.

Answer:

$10.8$