Question

kevin was asked to determine the length of side xz. his work is shown.
which error did kevin make?
he has the side lengths in the wrong place in the cosine ratio.
he multiplied both sides by the length of xz instead of dividing by xz.
he should have used the sine ratio.
he should have used the tangent ratio.

1. \\( \cos ( 34 ^ { \circ } ) = \frac { 18 } { x z } \\)
2. \\( ( x z ) \cos ( 34 ^ { \circ } ) = 18 \\)
3. \\( x z = \frac { 18 } { \cos ( 34 ^ { \circ } ) } \approx 21.7 \\)

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For angle \(X = 34^{\circ}\), the side \(YZ\) is opposite, \(XZ\) is adjacent and \(XY\) is hypotenuse.

Step2: Check the correct ratio

The correct ratio to find \(XZ\) (adjacent) when hypotenuse \(XY = 18\) is \(\cos(34^{\circ})=\frac{XZ}{18}\) (since \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)). But Kevin used \(\cos(34^{\circ})=\frac{18}{XZ}\) (he swapped adjacent and hypotenuse in the cosine ratio).

Answer:

He has the side lengths in the wrong place in the cosine ratio.