Question

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

Explanation:

Step1: Find side $r$ via Law of Cosines

$$r^2 = 5^2 + 11^2 - 2(5)(11)\cos(117^\circ)$$
Calculate each term:
$5^2=25$, $11^2=121$, $2(5)(11)=110$, $\cos(117^\circ)\approx-0.4540$
$$r^2 = 25 + 121 - 110(-0.4540) = 146 + 49.94 = 195.94$$
$$r = \sqrt{195.94} \approx 14.0$$

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

$$\frac{\sin\angle P}{11} = \frac{\sin(117^\circ)}{14.0}$$
$\sin(117^\circ)\approx0.8910$, so:
$$\sin\angle P = \frac{11 \times 0.8910}{14.0} \approx \frac{9.801}{14.0} \approx 0.7001$$
$$\angle P \approx \arcsin(0.7001) \approx 44.4^\circ$$

Step3: Find $\angle Q$ via angle sum

$$\angle Q = 180^\circ - 117^\circ - 44.4^\circ = 18.6^\circ$$

Answer:

$r \approx 14.0$, $\angle P \approx 44.4^\circ$, $\angle Q \approx 18.6^\circ$