Which of the following expressions can be used to find the measure of angle $B$ in a right triangle if the lengths of the sides opposite and adjacent to $B$ are known?

• A. $B=arctan\left(\frac{\mathrm{adjacent}}{\mathrm{opposite}}\right)$
• B. $B=arcsin\left(\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\right)$
• C. $B=arccos\left(\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\right)$
• D. $B=arctan\left(\frac{\mathrm{opposite}}{\mathrm{adjacent}}\right)$

Answer: (D)

Verified step by step guidance

1. Recall the definitions of the primary trigonometric ratios in a right triangle relative to angle \(B\): sine is \(\sin B = \frac{\text{opposite}}{\text{hypotenuse}}\), cosine is \(\cos B = \frac{\text{adjacent}}{\text{hypotenuse}}\), and tangent is \(\tan B = \frac{\text{opposite}}{\text{adjacent}}\).
2. Since the problem states that the lengths of the sides opposite and adjacent to angle \(B\) are known, focus on the tangent ratio, which directly relates these two sides.
3. To find the measure of angle \(B\) using the tangent ratio, use the inverse tangent function (arctangent), which gives \(B = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)\).
4. Check the other options: \(B = \arctan\left(\frac{\text{adjacent}}{\text{opposite}}\right)\) reverses the ratio and is incorrect; \(B = \arcsin\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\) and \(B = \arccos\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\) require the hypotenuse length, which is not given.
5. Therefore, the correct expression to find angle \(B\) when only the opposite and adjacent sides are known is \(B = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)\).