Question

consider right triangle $\triangle abc$ below.
which expressions are equivalent to $\sin(\angle b)$?
choose 2 answers:
a $\tan(\angle a)$
b $\dfrac{\text{length of side adjacent to }\angle a}{\text{length of hypotenuse}}$
c $\sin(\angle a)$
d $\dfrac{ac}{ab}$

Explanation:

Step1: Define $\sin(\angle B)$

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

Step2: Analyze Option B

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

Step3: Verify Option D

$\frac{AC}{AB}$ is exactly the expression for $\sin(\angle B)$ from Step1.

Step4: Eliminate other options

$\tan(\angle A)=\frac{BC}{AC}
eq\sin(\angle B)$, $\sin(\angle A)=\frac{BC}{AB}
eq\sin(\angle B)$

Answer:

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