Question

solve for x.
34
x°
32
84°
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 with sides \( a = 34 \), \( b = 32 \), angle \( B = 84^\circ \), and angle \( A = x^\circ \). Using the Law of Sines: \( \frac{\sin A}{a}=\frac{\sin B}{b} \)

Step2: Substitute values

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

Step3: Solve for \( \sin x \)

Multiply both sides by 34: \( \sin x=\frac{34\times\sin 84^\circ}{32} \)
Calculate \( \sin 84^\circ\approx0.9945 \), so \( \sin x=\frac{34\times0.9945}{32}=\frac{33.813}{32}\approx1.05665625 \) Wait, that can't be. Wait, maybe I mixed up the sides. Wait, the side opposite \( 84^\circ \) is 34? Wait no, the side with length 32 is opposite the \( 84^\circ \) angle? Wait, no, let's re - identify. Let's say: Let the angle opposite side 32 be \( 84^\circ \), so side \( a = 34 \) (opposite angle \( x \)), side \( b = 32 \) (opposite angle \( 84^\circ \)). So Law of Sines: \( \frac{\sin x}{34}=\frac{\sin 84^\circ}{32} \)
Wait, but \( \sin x=\frac{34\sin 84^\circ}{32}\approx\frac{34\times0.9945219}{32}=\frac{33.8137446}{32}\approx1.0566795 \). But the sine of an angle can't be more than 1. Oh, I must have mixed up the sides. So actually, the side opposite \( x \) is 32, and the side opposite \( 84^\circ \) is 34. Let's correct that. Let \( a = 32 \) (opposite \( x \)), \( b = 34 \) (opposite \( 84^\circ \)). Then Law of Sines: \( \frac{\sin x}{32}=\frac{\sin 84^\circ}{34} \)

Step4: Correct substitution and solve

Now, \( \sin x=\frac{32\times\sin 84^\circ}{34} \)
\( \sin 84^\circ\approx0.9945219 \)
\( \sin x=\frac{32\times0.9945219}{34}=\frac{31.8247008}{34}\approx0.9360206 \)

Step5: Find \( x \)

Take the inverse sine: \( x=\arcsin(0.9360206) \approx 69.51^\circ \) (rounded to the nearest hundredth)

Answer:

\( 69.51 \)