Question

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

Explanation:

Step1: Recall trigonometric - ratio definitions

In right - triangle \(ABC\) with right - angle at \(C\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle B\), \(\sin(B)=\frac{AC}{AB}=\frac{5}{13}\), and \(\cos(B)=\frac{BC}{AB}=\frac{12}{13}\). Also, for \(\angle A\), \(\sin(A)=\frac{BC}{AB}=\frac{12}{13}\) and \(\cos(A)=\frac{AC}{AB}=\frac{5}{13}\).

Step2: Identify equal trigonometric values

We know that \(\sin(B)=\cos(A)\) since \(\sin(B)=\frac{5}{13}\) and \(\cos(A)=\frac{5}{13}\), and \(\sin(A)=\cos(B)\) since \(\sin(A)=\frac{12}{13}\) and \(\cos(B)=\frac{12}{13}\).

Answer:

\(\sin(B)=\cos(A)\); \(\sin(A)=\cos(B)\)