Given a right triangle with angle $A$ and sides $a$ (opposite), $b$ (adjacent), and $c$ (hypotenuse), which of the following expressions can be used to find the measure of angle $A$? Select one correct option.

• A. ${cos}^{-1}\left(\frac{a}{c}\right)$
• B. ${sin}^{-1}\left(\frac{b}{c}\right)$
• C. ${cos}^{-1}\left(\frac{b}{c}\right)$
• D. ${sin}^{-1}\left(\frac{a}{c}\right)$

Answer: (D)

Verified step by step guidance

1. Identify the sides of the right triangle relative to angle \(A\): side \(a\) is opposite to angle \(A\), side \(b\) is adjacent to angle \(A\), and side \(c\) is the hypotenuse.
2. Recall the definitions of sine and cosine for angle \(A\): \(\sin(A) = \frac{a}{c}\) and \(\cos(A) = \frac{b}{c}\).
3. To find the measure of angle \(A\), use the inverse trigonometric functions (also called arc functions). Specifically, \(A = \sin^{-1}\left(\frac{a}{c}\right)\) or \(A = \cos^{-1}\left(\frac{b}{c}\right)\).
4. Check the given options and match them with the correct inverse function and ratio: the expression \(\cos^{-1}\left(\frac{b}{c}\right)\) correctly uses the adjacent side over hypotenuse for cosine, and \(\sin^{-1}\left(\frac{a}{c}\right)\) correctly uses the opposite side over hypotenuse for sine.
5. Conclude that the correct expressions to find angle \(A\) are \(A = \cos^{-1}\left(\frac{b}{c}\right)\) or \(A = \sin^{-1}\left(\frac{a}{c}\right)\), and any other combinations are incorrect.