Question

solve for x.
34
x°
84°
32
x = ?°
law of sines (\frac{sin a}{a}=\frac{sin b}{b}=\frac{sin c}{c})
round your answer to the nearest hundredth.

Explanation:

Step1: Identify sides and angles

Let's denote the triangle's sides: side \( a = 32 \), side \( b = 34 \), angle \( B = 84^\circ \), and we need to find angle \( A = x^\circ \). By the Law of Sines, \( \frac{\sin A}{a} = \frac{\sin B}{b} \).

Step2: Substitute values

Substitute \( a = 32 \), \( b = 34 \), \( B = 84^\circ \) into the formula: \( \frac{\sin x}{32} = \frac{\sin 84^\circ}{34} \).

Step3: Solve for \( \sin x \)

Multiply both sides by 32: \( \sin x = \frac{32 \times \sin 84^\circ}{34} \). Calculate \( \sin 84^\circ \approx 0.9945 \), so \( \sin x = \frac{32 \times 0.9945}{34} \approx \frac{31.824}{34} \approx 0.936 \).

Step4: Find \( x \)

Take the inverse sine: \( x = \arcsin(0.936) \approx 69.51^\circ \).

Answer:

\( 69.51 \)