Find the unknown sides of the right triangle (see illustration). Round your answer to the appropriate number of significant digits. $$b = 21.67 \mathrm {~km} , \angle B = 82.01 ^ { \circ }$$
A) a = 154.4 km, c = 21.46 km
B) a = 16.2 km, c = 27.04 km
C) a = 155.9 km, c = 154.4 km
D) a = 3.042 km, c = 21.88 km
E) a = 21.88 km, c = 3.042 km

Question

Find the unknown sides of the right triangle (see illustration). Round your answer to the appropriate number of significant digits. $$b = 21.67 \mathrm {~km} , \angle B = 82.01 ^ { \circ }$$ 
A) a = 154.4 km, c = 21.46 km 
B) a = 16.2 km, c = 27.04 km 
C) a = 155.9 km, c = 154.4 km 
D) a = 3.042 km, c = 21.88 km 
E) a = 21.88 km, c = 3.042 km

Quizplus · Accepted Answer

We can use the trigonometric ratios to solve for the unknown sides of the right triangle. We have:
$$\sin B = \frac{a}{c}$$ since $\sin B$ is opposite over hypotenuse
Solving for $a$, we get:
$$a = c \sin B = 21.67 \mathrm{~km} \times \sin(82.01 ^{\circ}) \approx 21.88 \mathrm{~km}$$
Next, we can use the Pythagorean theorem to solve for $c$:
$$c^2 = a^2 + b^2$$
$$c = \sqrt{a^2 + b^2} \approx \sqrt{(21.88 \mathrm{~km})^2 + (21.67 \mathrm{~km})^2} \approx  21.88 \mathrm{~km}$$
Therefore, the unknown sides are $a = 21.88 \mathrm{~km}$ and $c \approx 21.88 \mathrm{~km}$, and the best choice is D.

Find the Unknown Sides of the Right Triangle (See Illustration) b=21.67 km,∠B=82.01∘b = 21.67 \mathrm {~km} , \angle B = 82.01 ^ { \circ }b=21.67 km,∠B=82.01∘

Find the Unknown Sides of the Right Triangle (See Illustration) $b = 21.67 \mathrm {~km} , \angle B = 82.01 ^ { \circ }$