Geometry

Question Analysis

The main focus of this problem is to use inverse trigonometric functions to find the unknown angle  in a right - triangle. We need to determine the correct trigonometric ratio based on the given side lengths relative to angle  and then set up the appropriate inverse trigonometric equation.

 

Key Concept Explanation

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

, , .

The 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 adjacent to angle .

2. Recall the tangent function definition:

Since , substituting the values, we get .

3. Use the inverse tangent function:

To solve for , we apply the inverse tangent function. So,

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