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 (1)
The problem defines an angle x as the inverse sine of a ratio, specifically x = sin − 1 ( 8.3 5 ) .
In a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse.
Therefore, the triangle must have the side opposite to angle x with length 5 and the hypotenuse with length 8.3.
The measure of the unknown angle, x , is equal to the value of sin − 1 ( 8.3 5 ) .
Explanation
Understanding the problem The problem states that the measure of an unknown angle, x , is equal to sin − 1 ( 8.3 5 ) . This means we are looking for a right triangle where the ratio of the side opposite to angle x to the hypotenuse is 8.3 5 . In other words, sin ( x ) = 8.3 5 .
Relating sine to the triangle's sides 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, if x = sin − 1 ( 8.3 5 ) , then in the right triangle, the side opposite to angle x has a length of 5, and the hypotenuse has a length of 8.3.
Identifying the triangle Therefore, the triangle we are looking for is a right triangle where the side opposite to the angle x has length 5 and the hypotenuse has length 8.3.
Examples
Imagine you are building a ramp and you need it to rise at a certain angle. If you know the desired angle is x = sin − 1 ( 8.3 5 ) , it means for every 8.3 meters along the ramp (the hypotenuse), the ramp rises 5 meters in height (the opposite side). This ensures the ramp has the correct slope.
