A right triangle is shown below. Which ratio gives the cosine of $\angle B$?

• A. $\frac{\mathrm{opposite}}{\mathrm{adjacent}}$
• B. $\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$
• C. $\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$
• D.

Answer: (C)

Verified step by step guidance

1. Recall the definition of cosine in a right triangle: the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. This can be written as \(\cos(\angle B) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
2. Identify the sides relative to angle B in the triangle: the side next to angle B (but not the hypotenuse) is the adjacent side, the side opposite angle B is the opposite side, and the longest side is the hypotenuse.
3. Compare the given ratios to the cosine definition: \(\frac{\text{opposite}}{\text{adjacent}}\) and \(\frac{\text{opposite}}{\text{hypotenuse}}\) do not match the cosine ratio because cosine does not involve the opposite side in the numerator.
4. Confirm that the ratio \(\frac{\text{adjacent}}{\text{hypotenuse}}\) matches the cosine definition exactly, making it the correct ratio for \(\cos(\angle B)\).
5. Therefore, the ratio that gives the cosine of angle B is \(\frac{\text{adjacent}}{\text{hypotenuse}}\).