Question

determine the measure of angle θ to the nearest degree.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ for a triangle with sides $a, b, c$ and opposite angles $A, B, C$ respectively. In triangle $UVW$, we know side $UW = 30$ (opposite angle $V$), side $VW = 35.5$ (opposite angle $U = 93^\circ$), and we need to find angle $\theta$ (angle $W$). So we set up the proportion:
$\frac{35.5}{\sin 93^\circ} = \frac{30}{\sin \theta}$

Step2: Solve for $\sin \theta$

First, calculate $\sin 93^\circ \approx 0.9986$. Then rearrange the proportion to solve for $\sin \theta$:
$\sin \theta = \frac{30 \times \sin 93^\circ}{35.5}$
Substitute the value of $\sin 93^\circ$:
$\sin \theta = \frac{30 \times 0.9986}{35.5} \approx \frac{29.958}{35.5} \approx 0.8439$

Step3: Find the angle $\theta$

Take the inverse sine (arcsin) of $0.8439$ to find $\theta$:
$\theta \approx \arcsin(0.8439) \approx 57.5^\circ$
Round to the nearest degree: $\theta \approx 58^\circ$

Answer:

$58^\circ$