In which triangle is the measure of the unknown angle, [tex]$x$[/tex], equal to the value of [tex]$\sin ^{-1}\left(\frac{5}{8.3}\right) ?$[/tex]
Answer (2)
The problem requires finding a triangle where the angle x satisfies x = sin − 1 ( 8.3 5 ) .
This implies sin ( x ) = 8.3 5 , meaning the ratio of the opposite side to the hypotenuse in a right triangle is 8.3 5 .
Therefore, the correct triangle is a right triangle with the side opposite to angle x having length 5 and the hypotenuse having length 8.3.
The measure of the unknown angle, x , is equal to the value of sin − 1 ( 8.3 5 ) .
Explanation
Problem Analysis The problem states that the measure of the unknown angle, x , is equal to the value of sin − 1 ( 8.3 5 ) . This means we are looking for a triangle where the sine of the angle x is equal to 8.3 5 . In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Therefore, we need to identify a triangle where the ratio of the opposite side to the hypotenuse is 8.3 5 .
Finding the Correct Triangle We are looking for a right triangle where the ratio of the side opposite to angle x to the hypotenuse is equal to 8.3 5 . This means that if we have a right triangle with angle x , the side opposite to x has length 5 and the hypotenuse has length 8.3.
Conclusion Therefore, the triangle in which the measure of the unknown angle, x , is equal to the value of sin − 1 ( 8.3 5 ) is a right triangle where the side opposite to angle x has length 5 and the hypotenuse has length 8.3.
Examples
Imagine you're designing a ramp for a skateboard park. You need the ramp to have a specific angle of elevation for safety and usability. If you know the desired ratio of the ramp's height to its length (the hypotenuse), you can use the arcsine function to determine the exact angle needed. For instance, if you want the height to be 5 feet and the length to be 8.3 feet, the angle of elevation would be sin − 1 ( 8.3 5 ) , ensuring the ramp is built to the correct specifications.
The angle x equals sin − 1 ( 8.3 5 ) in a right triangle where the opposite side is 5 and the hypotenuse is 8.3. This means the triangle is specifically defined by these side lengths. The sine ratio for this triangle fits the properties attributed to the angle x .
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