Question

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

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle \(\triangle ABC\) with \(\angle B = 90^{\circ}\), \(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{b}\) and \(\cos(C)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{a}{b}\). Also, \(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{c}{b}\) and \(\sin(C)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{c}{b}\).

Step2: Compare the ratios

Since \(\sin(A)=\frac{a}{b}\) and \(\cos(C)=\frac{a}{b}\), we have \(\sin(A)=\cos(C)\). Since \(\cos(A)=\frac{c}{b}\) and \(\sin(C)=\frac{c}{b}\), we have \(\cos(A)=\sin(C)\).

Answer:

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