Find the exact value, if any, of the following composite function. Do not use a calculator.
[tex]$\sin ^{-1}\left(\sin \frac{11 \pi}{8}\right)$[/tex]
Select the correct choice and, if necessary, fill in the answer box within your choice.
A. [tex]$\sin ^{-1}\left(\sin \frac{11 \pi}{8}\right)=[/tex] $\square$ (Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed. Use integers or fractions for any numbers in the expression.)
B. It is not defined.
Answer (2)
Recognize that 8 11 π is outside the range of sin − 1 ( x ) , which is [ − 2 π , 2 π ] .
Use the property sin ( π + x ) = − sin ( x ) to rewrite sin ( 8 11 π ) as − sin ( 8 3 π ) .
Apply the property sin ( − x ) = − sin ( x ) to find an equivalent angle within the desired range: − 8 3 π .
Conclude that sin − 1 ( sin ( 8 11 π )) = − 8 3 π .
Explanation
Understanding the Problem We are asked to find the exact value of the composite function sin − 1 ( sin ( 8 11 π )) . The key to solving this problem is understanding the range of the inverse sine function.
Range of Inverse Sine The range of the inverse sine function, sin − 1 ( x ) , is [ − 2 π , 2 π ] . This means that the output of sin − 1 ( x ) must be within this interval. The value 8 11 π is not within this range, so we need to find an angle within the range that has the same sine value.
Using Trigonometric Properties The angle 8 11 π is in the third quadrant, where sine is negative. To find an equivalent angle within the range of the inverse sine function, we can use the property that sin ( π + x ) = − sin ( x ) . Thus, sin ( 8 11 π ) = sin ( π + 8 3 π ) = − sin ( 8 3 π ) .
Finding the Equivalent Angle We want to find an angle θ in the interval [ − 2 π , 2 π ] such that sin ( θ ) = − sin ( 8 3 π ) . Since sin ( − x ) = − sin ( x ) , we can choose θ = − 8 3 π . This angle is indeed in the interval [ − 2 π , 2 π ] .
Final Calculation Therefore, sin − 1 ( sin ( 8 11 π )) = − 8 3 π .
Final Answer The exact value of the composite function sin − 1 ( sin ( 8 11 π )) is − 8 3 π .
Examples
Imagine you are designing a robotic arm that needs to reach a specific angle. The arm's control system uses trigonometric functions to determine the joint angles. If the required angle is outside the standard range of the inverse sine function, you need to find an equivalent angle within that range to ensure the arm moves correctly. This problem demonstrates how to find such equivalent angles, ensuring the robotic arm accurately reaches its target position.
The exact value of the composite function sin − 1 ( sin 8 11 π ) is − 8 3 π . It is derived by using properties of the sine function and the defined range of the inverse sine function.
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