Find the exact value, if any, of the following composition: [tex]$\sin ^{-1}\left(\sin \frac{3 \pi}{5}\right)$[/tex]
Answer (2)
The exact value of sin − 1 ( sin 5 3 π ) is 5 2 π because it reflects the equivalent angle within the range of the inverse sine function after applying the identity for sine.
This process concludes that sin ( 5 3 π ) equals sin ( 5 2 π ) , ensuring we find an equivalent angle in the correct range.
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Recognize that 5 3 π is not within the range of sin − 1 ( x ) , which is [ − 2 π , 2 π ] .
Use the identity sin ( x ) = sin ( π − x ) to find an equivalent angle: sin ( 5 3 π ) = sin ( π − 5 3 π ) = sin ( 5 2 π ) .
Verify that 5 2 π is within the range of sin − 1 ( x ) .
Conclude that sin − 1 ( sin ( 5 3 π )) = 5 2 π , so the final answer is 5 2 π .
Explanation
Understanding the Problem We are asked to find the exact value of the composite function sin − 1 ( sin ( 5 3 π )) . This problem involves understanding the range of the inverse sine function and using trigonometric identities to simplify the expression.
Checking the Range The range of the inverse sine function, sin − 1 ( x ) , is [ − 2 π , 2 π ] . This means that the output of the sin − 1 function must be within this interval. We need to check if 5 3 π lies within this range. We know that π ≈ 3.14 , so 5 3 π ≈ 5 3 × 3.14 ≈ 1.88 . Since 2 π ≈ 2 3.14 ≈ 1.57 , we see that 5 3 π is not in the range of the inverse sine function.
Using Trigonometric Identities Since 5 3 π is not in the range of sin − 1 ( x ) , we need to find an angle θ in the interval [ − 2 π , 2 π ] such that sin ( θ ) = sin ( 5 3 π ) . We can use the identity sin ( x ) = sin ( π − x ) . Therefore, sin ( 5 3 π ) = sin ( π − 5 3 π ) = sin ( 5 2 π ) .
Finding the Equivalent Angle Now we need to check if 5 2 π is in the range of sin − 1 ( x ) . We have 5 2 π ≈ 5 2 × 3.14 ≈ 1.26 . Since − 2 π ≤ 5 2 π ≤ 2 π , we have that 5 2 π is in the range of sin − 1 ( x ) . Therefore, sin − 1 ( sin ( 5 3 π )) = sin − 1 ( sin ( 5 2 π )) = 5 2 π .
Final Answer The exact value of sin − 1 ( sin ( 5 3 π )) is 5 2 π .
Examples
Imagine you're designing a robotic arm that needs to reach a specific angle. The arm's control system uses inverse trigonometric functions to determine the joint angles required to achieve the desired position. If the desired angle is outside the principal 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 that equivalent angle using trigonometric identities, ensuring the robotic arm functions as intended.
