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 sine of an angle x is equal to 8.3 5 .
This is equivalent to finding a right triangle where the ratio of the side opposite to angle x to the hypotenuse is 8.3 5 .
The triangle we are looking for has the side opposite to the unknown angle x equal to 5 and the hypotenuse equal to 8.3.
Therefore, the triangle in question is the one where sin ( x ) = 8.3 5 . sin ( x ) = 8.3 5
Explanation
Problem Analysis The problem states that we need to find a triangle where the unknown angle x satisfies the equation x = sin − 1 ( 8.3 5 ) . This is equivalent to finding a triangle where sin ( x ) = 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 are looking for a right triangle where the ratio of the opposite side to the hypotenuse is 8.3 5 .
Solution Strategy To find the triangle, we need to examine each triangle and check if the ratio of the side opposite to angle x to the hypotenuse is equal to 8.3 5 . This means we are looking for a triangle where the opposite side is 5 and the hypotenuse is 8.3.
Finding the Triangle The triangle in which the measure of the unknown angle, x , is equal to the value of sin − 1 ( 8.3 5 ) is the triangle where the side opposite to angle x is 5 and the hypotenuse is 8.3. Therefore, sin ( x ) = 8.3 5 .
Final Answer The triangle we are looking for has the side opposite to the unknown angle x equal to 5 and the hypotenuse equal to 8.3.
Examples
Imagine you are building a ramp and you want it to have an angle of elevation equal to x . If you know that sin ( x ) = 8.3 5 , it means that for every 8.3 meters of ramp length (hypotenuse), the ramp rises 5 meters in height (opposite side). This ensures the ramp has the desired angle.
The measure of the unknown angle x is given by sin − 1 ( 8.3 5 ) , indicating that in a right triangle, the ratio of the opposite side to the hypotenuse is 8.3 5 . Thus, the triangle we are looking for has an opposite side length of 5 and a hypotenuse length of 8.3.
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