Geometry

Question Analysis

The main focus is on using inverse trigonometric functions to find the unknown angle  in a right - triangle. We must identify the correct trigonometric ratio based on the side lengths relative to angle  and then set up the appropriate inverse trig equation.

 

Key Concept Explanation

1. In a right - triangle, for an acute angle :

, , .

Inverse trigonometric functions , ,  are used to find the angle when the ratio of the sides is known. For example, if , then ; if , then ; if , then .

 

Step - by - Step Solution

1. Identify the sides relative to angle :

The side of length  is opposite to angle , and the side of length  is the hypotenuse.

2. Recall the sine function definition:

Since , substituting the values, we get