In which triangle is the value of $x$ equal to $\tan ^{-1}\left[\frac{3.1}{5.2}\right]$? (Images may not be drawn to scale.)
Answer (2)
The problem states that tan ( x ) = 5.2 3.1 .
Triangle A has the opposite side as 3.1 and the adjacent side as 5.2, so tan ( x ) = 5.2 3.1 .
Triangle B has the opposite side as 5.2 and the adjacent side as 3.1, so tan ( x ) = 3.1 5.2 .
Triangle C has the opposite side as 3.1 and the adjacent side as 6.2, so tan ( x ) = 6.2 3.1 .
Therefore, the value of x is equal to tan − 1 [ 5.2 3.1 ] in T r ian g l e A .
Explanation
Understanding the problem We are given the expression x = tan − 1 [ 5.2 3.1 ] . This means that tan ( x ) = 5.2 3.1 . We need to find the triangle where the ratio of the side opposite to angle x to the side adjacent to angle x is equal to 5.2 3.1 .
Analyzing each triangle Let's examine each triangle:
Triangle A: The side opposite to angle x is 3.1, and the side adjacent to angle x is 5.2. Therefore, tan ( x ) = 5.2 3.1 .
Triangle B: The side opposite to angle x is 5.2, and the side adjacent to angle x is 3.1. Therefore, tan ( x ) = 3.1 5.2 .
Triangle C: The side opposite to angle x is 3.1, and the side adjacent to angle x is 6.2. Therefore, tan ( x ) = 6.2 3.1 .
Finding the matching triangle Comparing the values of tan ( x ) for each triangle with the given value 5.2 3.1 , we find that Triangle A has tan ( x ) = 5.2 3.1 , which matches the given condition.
Final Answer Therefore, the value of x is equal to tan − 1 [ 5.2 3.1 ] in Triangle A.
Examples
Imagine you're building a ramp and need to determine the angle of elevation. If you know the rise (vertical height) is 3.1 meters and the run (horizontal distance) is 5.2 meters, you can use the arctangent function to find the angle. This ensures your ramp has the correct slope for safety and ease of use. The angle x would be tan − 1 ( 5.2 3.1 ) , which is approximately 30.79 degrees.
The value of x is equal to tan − 1 [ 5.2 3.1 ] in Triangle A, where the opposite side is 3.1 and the adjacent side is 5.2.
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