Question

in the triangle below, which is equivalent to sina?

a

b

c

sinc

sinb

cosa

cosc

Explanation:

Step1: Recall trigonometric ratios in right triangles

In a right triangle, for an acute angle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Also, in right triangle \(ABC\) with right angle at \(B\), \(\angle A + \angle C=90^\circ\), so \(\angle A = 90^\circ-\angle C\).

Step2: Express \(\sin A\) and \(\cos C\)

For \(\angle A\): \(\sin A=\frac{BC}{\text{hypotenuse}(AC)}\) (opposite side to \(A\) is \(BC\), hypotenuse is \(AC\)).
For \(\angle C\): \(\cos C=\frac{BC}{\text{hypotenuse}(AC)}\) (adjacent side to \(C\) is \(BC\), hypotenuse is \(AC\)).
So \(\sin A = \cos C\).

Answer:

D. \(\cos C\) (assuming the last option is D. \(\cos C\))