Question

solve the triangle using the law of sines. (assume b = 8, ∠a = 50°, and ∠c = 120°. round the lengths to two decimal places.) a = c = ∠b =

Explanation:

Step1: Find angle B

The sum of angles in a triangle is 180°. So, $\angle B=180^{\circ}-\angle A - \angle C$.
$\angle B = 180^{\circ}-50^{\circ}-120^{\circ}=10^{\circ}$

Step2: Use Law of Sines to find side a

The Law of Sines states $\frac{a}{\sin A}=\frac{b}{\sin B}$. We know $b = 8$, $\angle A=50^{\circ}$, $\angle B = 10^{\circ}$. So, $a=\frac{b\sin A}{\sin B}$.
$a=\frac{8\times\sin50^{\circ}}{\sin10^{\circ}}\approx\frac{8\times0.7660}{0.1736}\approx34.95$

Step3: Use Law of Sines to find side c

Using the Law of Sines $\frac{c}{\sin C}=\frac{b}{\sin B}$. We know $b = 8$, $\angle C=120^{\circ}$, $\angle B = 10^{\circ}$. So, $c=\frac{b\sin C}{\sin B}$.
$c=\frac{8\times\sin120^{\circ}}{\sin10^{\circ}}\approx\frac{8\times0.8660}{0.1736}\approx39.78$

Answer:

$a\approx34.95$
$c\approx39.78$
$\angle B = 10^{\circ}$