Given a right triangle with angle $A$, which of the following expressions can be used to find the measure of angle $A$? Select three options.

• A. ${\cos }^{-1}\frac{\mathrm{opposite}}{\mathrm{adjacent}}$
• B. ${\sin }^{-1}\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$
• C. ${\tan }^{-1}\frac{\mathrm{opposite}}{\mathrm{adjacent}}$
• D. ${\cos }^{-1}\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$

Answer: (D)

Verified step by step guidance

1. Identify the sides of the right triangle relative to angle A: the opposite side is opposite angle A, the adjacent side is next to angle A (but not the hypotenuse), and the hypotenuse is the longest side opposite the right angle.
2. Recall the definitions of the primary trigonometric ratios for angle A: \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan A = \frac{\text{opposite}}{\text{adjacent}}\).
3. To find the measure of angle A from a ratio, use the inverse trigonometric functions: \(A = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\), \(A = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\), or \(A = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\).
4. Evaluate each given expression to see if it matches one of these inverse trig functions with the correct ratio of sides relative to angle A.
5. Select the three expressions that correctly use inverse sine, inverse cosine, or inverse tangent with the appropriate side ratios to find angle A.