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

$$w^2 = 9^2 + 11^2 - 2(9)(11)\cos(103^\circ)$$
Calculate each term:
$9^2=81$, $11^2=121$, $2(9)(11)=198$, $\cos(103^\circ)\approx-0.2250$
$$w^2 = 81 + 121 - 198(-0.2250) = 202 + 44.55 = 246.55$$
$$w = \sqrt{246.55} \approx 15.7$$

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

$$\frac{\sin\angle X}{9} = \frac{\sin(103^\circ)}{15.7}$$
$\sin(103^\circ)\approx0.9744$
$$\sin\angle X = \frac{9\times0.9744}{15.7} \approx \frac{8.7696}{15.7} \approx 0.5586$$
$$\angle X = \arcsin(0.5586) \approx 33.9^\circ$$

Step3: Find $\angle Y$ via angle sum

$$\angle Y = 180^\circ - 103^\circ - 33.9^\circ = 43.1^\circ$$

Answer:

$w \approx 15.7$, $\angle X \approx 33.9^\circ$, $\angle Y \approx 43.1^\circ$