Question

which relationship in the triangle must be true?
a
c
b
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 identities

In a right - triangle \(ABC\) with \(\angle C = 90^{\circ}\), \(\angle A+\angle B=90^{\circ}\), so \(\angle A = 90^{\circ}-\angle B\).
We know the co - function identity: \(\sin(x)=\cos(90^{\circ}-x)\) and \(\cos(x)=\sin(90^{\circ}-x)\) for any angle \(x\) such that \(0^{\circ}

Step2: Analyze each option

• Option 1: \(\sin(B)=\sin(A)\). Since \(A = 90^{\circ}-B\), \(\sin(B)

eq\sin(90^{\circ}-B)\) in general (except when \(B = 45^{\circ}\)).

• Option 2: \(\sin(B)=\cos(90 - B)\). By the co - function identity \(\sin(x)=\cos(90^{\circ}-x)\), when \(x = B\), this is true.
• Option 3: \(\cos(B)=\sin(180 - B)\). We know that \(\sin(180 - B)=\sin B

eq\cos B\) in general.

• Option 4: \(\cos(B)=\cos(A)\). Since \(A = 90^{\circ}-B\), \(\cos(B)

eq\cos(90^{\circ}-B)\) in general (except when \(B = 45^{\circ}\)).

Answer:

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