Given a right triangle with angle $A$, opposite side $a$, adjacent side $b$, and hypotenuse $c$, which expression can be used to find $\angle A$?

• A. $\angle A=\arctan \left(\frac{a}{b}\right)$
• B. $\angle A=\arccos \left(\frac{a}{b}\right)$
• C. $\angle A=\arccos \left(\frac{b}{c}\right)$
• D. $\angle A=\arcsin \left(\frac{a}{c}\right)$

Answer: (D)

Verified step by step guidance

1. Identify the sides relative to angle A in the right triangle: side a is opposite to angle A, side b is adjacent to angle A, and side c is the hypotenuse.
2. Recall the primary trigonometric ratios for a right triangle: sine, cosine, and tangent, defined as follows: \(\sin(A) = \frac{a}{c}\), \(\cos(A) = \frac{b}{c}\), and \(\tan(A) = \frac{a}{b}\).
3. To find angle A given the sides, use the inverse trigonometric functions (arcsin, arccos, arctan) applied to the appropriate ratio of sides.
4. Match each inverse function with the correct ratio: \(A = \arcsin\left(\frac{a}{c}\right)\), \(A = \arccos\left(\frac{b}{c}\right)\), and \(A = \arctan\left(\frac{a}{b}\right)\).
5. Choose the expression that correctly corresponds to the given sides and the angle A you want to find, based on the ratio of opposite, adjacent, and hypotenuse sides.