Find the exact value of the following expression.
[tex]$\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)$[/tex]
Select the correct choice below and, if necessary, fill in the answer.
A. [tex]$\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)=[/tex] $\square$
(Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed.)
Answer (1)
The problem asks to find the exact value of sin − 1 ( 2 2 ) .
We need to find an angle θ in the interval [ − 2 π , 2 π ] such that sin ( θ ) = 2 2 .
Recall that sin ( 4 π ) = 2 2 .
Therefore, sin − 1 ( 2 2 ) = 4 π .
Explanation
Understanding the Problem We are asked to find the exact value of sin − 1 ( 2 2 ) . This means we need to find an angle θ such that sin ( θ ) = 2 2 , and − 2 π ≤ θ ≤ 2 π .
Finding the Angle Recall the unit circle and the values of sine for common angles. We know that sin ( 4 π ) = 2 2 . Since 4 π is in the interval [ − 2 π , 2 π ] , it is the correct angle.
Final Answer Therefore, sin − 1 ( 2 2 ) = 4 π .
Examples
Imagine you are designing a ramp for a skateboard park. You want the ramp to have an angle such that the height gained is 2 2 times the length of the ramp. Using the inverse sine function, you can calculate the exact angle needed for the ramp, ensuring it meets safety and performance requirements. This problem demonstrates how trigonometric functions and their inverses are used in real-world design and engineering applications.
