Question

kevin was asked to determine the length of side xz. his work is shown.

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

which error did kevin make?
\bigcirc he has the side lengths in the wrong place in the cosine ratio.
\bigcirc he multiplied both sides by the length of xz instead of dividing by xz.
\bigcirc he should have used the sine ratio.
\bigcirc he should have used the tangent ratio.

Explanation:

Step1: Recall cosine definition

For angle $34^\circ$ at $X$, $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$.
Adjacent side to $\angle X$ is $XZ$, hypotenuse is $XY=18$. So correct ratio: $\cos(34^\circ)=\frac{XZ}{18}$

Step2: Compare to Kevin's work

Kevin wrote $\cos(34^\circ)=\frac{18}{XZ}$, swapping adjacent and hypotenuse.

Step3: Evaluate error options

Kevin's mistake is misplacing sides in cosine ratio, not ratio choice or algebra error.

Answer:

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