Question

consider right triangle $\triangle abc$ below.

which expressions are equivalent to $\sin(\angle b)$?
choose 2 answers:
a $\tan(\angle a)$
b $\frac{\text{length of side adjacent to }\angle a}{\text{length of hypotenuse}}$
c $\sin(\angle a)$
d $\frac{ac}{ab}$

Explanation:

Step1: Define sin(∠B) in △ABC

In right triangle $\triangle ABC$ (right angle at $C$), $\sin(\angle B) = \frac{\text{opposite side to } \angle B}{\text{hypotenuse}} = \frac{AC}{AB}$

Step2: Analyze Option A

$\tan(\angle A) = \frac{\text{opposite side to } \angle A}{\text{adjacent side to } \angle A} = \frac{BC}{AC}$. This does not equal $\sin(\angle B)$.

Step3: Analyze Option B

$\frac{\text{length of side adjacent to } \angle A}{\text{length of hypotenuse}} = \frac{AC}{AB}$, which matches $\sin(\angle B)$.

Step4: Analyze Option C

$\sin(\angle A) = \frac{\text{opposite side to } \angle A}{\text{hypotenuse}} = \frac{BC}{AB}$. This does not equal $\sin(\angle B)$.

Step5: Analyze Option D

$\frac{AC}{AB}$ is exactly the definition of $\sin(\angle B)$ we found in Step1.

Answer:

B. $\frac{\text{length of side adjacent to } \angle A}{\text{length of hypotenuse}}$
D. $\frac{AC}{AB}$