Question

hich relationship in the triangle must be true?

Triangle A B C is shown. Angle A C B is a right angle. The length of side C B is a, the length of side A C is b, and the length of side A B is c.

sin(B) = sin(A)
sin(B) = cos(90 – B)
cos(B) = sin(180 – B)
cos(B) = cos(A)

Answer 1

Only sin(B) = cos(90 − B) must be true.

Reason: C = 90°, so A + B = 90° → A = 90° − B. By the co-function identity cos(90° − B) = sin(B), so sin(B) = cos(90 − B).

The others are not generally true:
- sin(B) = sin(A) would require sin(B) = sin(90° − B) = cos(B), which only holds when B = 45°.
- cos(B) = sin(180° − B) becomes cos(B) = sin(B), again only when B = 45°.
- cos(B) = cos(A) becomes cos(B) = cos(90° − B) = sin(B), same restriction.