Question

consider △abc.
what are the angles that make the trigonometric statements true?
sin(□) = cos(b)
sin(b) = cos(□)

Explanation:

Step1: Recall co-function identity

For any acute angle $\theta$, $\sin(\theta) = \cos(90^\circ - \theta)$ and $\cos(\theta) = \sin(90^\circ - \theta)$. In right $\triangle ABC$, $\angle A + \angle B = 90^\circ$.

Step2: Solve first statement

$\sin(\text{?}) = \cos(B)$. Using the identity, $\cos(B) = \sin(90^\circ - B)$. Since $90^\circ - B = \angle A$, this means $\sin(A) = \cos(B)$.

Step3: Solve second statement

$\sin(B) = \cos(\text{?})$. Using the identity, $\sin(B) = \cos(90^\circ - B)$. Since $90^\circ - B = \angle A$, this means $\sin(B) = \cos(A)$.

Answer:

$\sin(\boldsymbol{A}) = \cos(B)$
$\sin(B) = \cos(\boldsymbol{A})$