Question

solving for side lengths of right triangles
which relationship in the triangle must be true?
sin (b) = sin (a)
cos (b) = cos (a)
cos (b) = sin (180 - b)
sin (b) = cos (90 - b)

Explanation:

Step1: Recall trigonometric co - function identity

The co - function identity states that $\sin\theta=\cos(90^{\circ}-\theta)$ for any angle $\theta$. In a right - triangle, if one acute angle is $B$, the other acute angle $A = 90^{\circ}-B$ (since the sum of the two acute angles in a right - triangle is $90^{\circ}$).

Step2: Analyze each option

• Option $\sin(B)=\sin(A)$: In a right - triangle, if $A$ and $B$ are acute angles ($A + B=90^{\circ}$), $\sin(A)=\frac{a}{c}$ and $\sin(B)=\frac{b}{c}$, and generally $a

eq b$, so $\sin(B)
eq\sin(A)$.

• Option $\cos(B)=\cos(A)$: $\cos(A)=\frac{b}{c}$ and $\cos(B)=\frac{a}{c}$, and generally $a

eq b$, so $\cos(B)
eq\cos(A)$.

• Option $\cos(B)=\sin(180 - B)$: We know that $\sin(180 - B)=\sin B$, and $\cos B

eq\sin B$ for acute non - $45^{\circ}$ angles in a right - triangle.

• Option $\sin(B)=\cos(90 - B)$: By the co - function identity, this is always true for any angle $B$.

Answer:

$\sin(B)=\cos(90 - B)$