Question

kevin was asked to determine the length of side xz. his work is shown.
triangle image with x, y, z; xy=18, angle at x is 34°, right angle at z

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

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.

Explanation:

In a right - triangle, the cosine of an angle is defined as the adjacent side divided by the hypotenuse. For angle \(X = 34^{\circ}\), the adjacent side to angle \(X\) is \(XZ\) and the hypotenuse is \(XY = 18\). So the correct cosine ratio should be \(\cos(34^{\circ})=\frac{XZ}{18}\), but Kevin used \(\cos(34^{\circ})=\frac{18}{XZ}\), which means he put the side lengths in the wrong place in the cosine ratio.

• Option 2: When solving \(\cos(34^{\circ})=\frac{XZ}{18}\) for \(XZ\), we multiply both sides by 18, not \(XZ\). Kevin's step 2 is wrong in the operation, but the main error in the ratio setup is the first option.
• Option 3: The sine ratio is \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), and for this problem, the cosine ratio (if set up correctly) is appropriate, so using sine is not the error.
• Option 4: The tangent ratio is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), and the cosine ratio (when set up correctly) is the right approach here, so using tangent is not the error.

Answer:

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