Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Find side $w$ via Law of Cosines

The Law of Cosines states $w^2 = a^2 + b^2 - 2ab\cos(C)$. Here, $a=7$, $b=14$, $C=30^\circ$.

$$\begin{align*} w^2 &= 7^2 + 14^2 - 2(7)(14)\cos(30^\circ)\\ w^2 &= 49 + 196 - 196\cdot\frac{\sqrt{3}}{2}\\ w^2 &= 245 - 98\sqrt{3}\\ w^2 &\approx 245 - 169.7\\ w^2 &\approx 75.3\\ w &\approx \sqrt{75.3} \approx 8.7 \end{align*}$$

Step2: Find $\angle X$ via Law of Sines

The Law of Sines states $\frac{\sin(X)}{7} = \frac{\sin(30^\circ)}{w}$.

$$\begin{align*} \sin(X) &= \frac{7\sin(30^\circ)}{w}\\ \sin(X) &\approx \frac{7\cdot0.5}{8.7}\\ \sin(X) &\approx \frac{3.5}{8.7} \approx 0.402\\ \angle X &\approx \arcsin(0.402) \approx 23.7^\circ \end{align*}$$

Step3: Find $\angle V$ via angle sum property

Triangle angles sum to $180^\circ$.

$$\begin{align*} \angle V &= 180^\circ - 30^\circ - 23.7^\circ\\ \angle V &= 126.3^\circ \end{align*}$$

Answer:

$w \approx 8.7$, $\angle X \approx 23.7^\circ$, $\angle V \approx 126.3^\circ$