Question

17
in the figure above, \\( \sin 52 ^ { \circ } = \frac { 17 } { c } \\).
based on the figure, which of the following equations is also true?
a \\( \sin 38 ^ { \circ } = \frac { c } { 17 } \\)
b \\( \cos 38 ^ { \circ } = \frac { 17 } { c } \\)
c \\( \cos 52 ^ { \circ } = \frac { 17 } { c } \\)
d \\( \tan 52 ^ { \circ } = \frac { c } { 17 } \\)

Explanation:

Step1: Find the third - angle of the right - triangle

In a right - triangle, the sum of the two non - right angles is \(90^{\circ}\). Let the third angle be \(\theta\). Given one non - right angle is \(52^{\circ}\), then \(\theta=90^{\circ}-52^{\circ} = 38^{\circ}\).

Step2: Recall the co - function identities

We know the co - function identity \(\sin\alpha=\cos(90^{\circ}-\alpha)\) and \(\cos\alpha=\sin(90^{\circ}-\alpha)\). Also, for a right - triangle with hypotenuse \(c\), adjacent side \(a\), and opposite side \(o\), \(\sin\theta=\frac{o}{c}\), \(\cos\theta=\frac{a}{c}\), and \(\tan\theta=\frac{o}{a}\).
Since \(\sin52^{\circ}=\frac{17}{c}\) (opposite side to \(52^{\circ}\) is \(17\)), and \(\cos38^{\circ}=\sin(90^{\circ} - 38^{\circ})=\sin52^{\circ}\)

Answer:

B. \(\cos38^{\circ}=\frac{17}{c}\)