Question

which relationship in the triangle must be true? a b c c a b sin(b) = sin(a) sin(b) = cos(90 - b) cos(b) = sin(180 - b) cos(b) = cos(a)

Explanation:

Step1: Recall trigonometric co - function identity

The co - function identity states that $\sin(x)=\cos(90 - x)$ for any angle $x$.

Step2: Analyze each option

• Option 1: $\sin(B)=\frac{b}{c}$ and $\sin(A)=\frac{a}{c}$, in a non - isosceles right - triangle $a

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

• Option 2: By the co - function identity, for any angle $B$, $\sin(B)=\cos(90 - B)$.
• Option 3: $\sin(180 - B)=\sin(B)$ and $\cos(B)

eq\sin(B)$ (except for some special angles), so $\cos(B)
eq\sin(180 - B)$.

• Option 4: $\cos(B)=\frac{a}{c}$ and $\cos(A)=\frac{b}{c}$, in a non - isosceles right - triangle $a

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

Answer:

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