In a right triangle, angle $B$ is an acute angle. Which expression correctly gives $\cos B$ in terms of the side lengths relative to angle $B$?

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

Answer: (C)

Verified step by step guidance

1. Recall the definition of cosine in a right triangle: for an acute angle \( B \), \( \cos(B) \) is the ratio of the length of the side adjacent to angle \( B \) to the length of the hypotenuse.
2. Identify the sides relative to angle \( B \): the 'adjacent' side is the one next to angle \( B \) (but not the hypotenuse), the 'opposite' side is across from angle \( B \), and the 'hypotenuse' is the longest side opposite the right angle.
3. Write the cosine ratio using these side lengths: \( \cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
4. Compare this expression to the given options to determine which matches the correct cosine definition.
5. Conclude that the correct expression for \( \cos(B) \) is the ratio of the adjacent side over the hypotenuse.