Find the exact value of the following expression.
$\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)$
Answer (1)
Recognize that the problem asks for the angle whose sine is 2 2 .
Recall the common trigonometric values and identify that sin ( 4 π ) = 2 2 .
Confirm that 4 π lies within the range of the inverse sine function, which is [ − 2 π , 2 π ] .
State the final answer: 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 We know that sin ( 4 π ) = 2 2 . Since 4 π is in the range of the inverse sine function ( [ − 2 π , 2 π ] ), we have sin − 1 ( 2 2 ) = 4 π .
Final Answer Therefore, the exact value of the expression is 4 π .
Examples
Imagine you're designing a ramp for a skateboard park. You want the ramp to have an angle such that the ratio of the opposite side (height) to the hypotenuse (length of the ramp) is 2 2 . Using the inverse sine function, you can determine that the angle of the ramp should be 4 π radians, or 45 degrees. This ensures the ramp has the desired steepness for skateboarders.
