Question

in the given figure, (\triangle abc) is a right triangle. what is true about (\triangle abc)? a. (sin(a)=cos(c)) and (cos(a)=cos(c)) b. (sin(a)=cos(a)) and (sin(c)=cos(c)) c. (sin(a)=sin(c)) and (cos(a)=cos(c)) d. (sin(a)=cos(c)) and (cos(a)=sin(c))

Explanation:

Step1: Recall trigonometric ratio definitions

In a right - triangle \(\triangle ABC\) with right - angle at \(B\), \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{b}\), \(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{c}{b}\), \(\sin(C)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{c}{b}\), \(\cos(C)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{a}{b}\)

Step2: Compare the ratios

We can see that \(\sin(A)=\frac{a}{b}\) and \(\cos(C)=\frac{a}{b}\), so \(\sin(A)=\cos(C)\). Also, \(\cos(A)=\frac{c}{b}\) and \(\sin(C)=\frac{c}{b}\), so \(\cos(A)=\sin(C)\)

Answer:

D. \(\sin(A)=\cos(C)\) and \(\cos(A)=\sin(C)\)