Find the exact value, if any, of the following composite function. Do not use
[tex]$\tan ^{-1}\left[\tan \left(-\frac{6 \pi}{19}\right)\right]$[/tex]
Select the correct choice below and, if necessary, fill in the answer box within
A. [tex]$\tan ^{-1}\left[\tan \left(-\frac{6 \pi}{19}\right)\right]$[/tex]
(Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed.
Answer (2)
The problem asks to evaluate the composite function tan − 1 ( tan ( − 19 6 π )) .
The range of tan − 1 ( x ) is ( − 2 π , 2 π ) .
Since − 19 6 π lies within the range ( − 2 π , 2 π ) , the composite function simplifies to − 19 6 π .
Therefore, the exact value is − 19 6 π .
Explanation
Understanding the problem We are asked to find the exact value of the composite function tan − 1 ( tan ( − 19 6 π )) .
Range of inverse tangent The range of the inverse tangent function, tan − 1 ( x ) , is ( − 2 π , 2 π ) . This means that the output of the tan − 1 function must lie within this interval.
Checking the interval We need to check if − 19 6 π lies within the interval ( − 2 π , 2 π ) . From the tool, we know that − 2 π < − 19 6 π < 2 π is True.
Final Answer Since − 19 6 π is within the interval ( − 2 π , 2 π ) , the exact value of the composite function is simply − 19 6 π . Therefore, tan − 1 ( tan ( − 19 6 π )) = − 19 6 π .
Examples
Imagine you're designing a satellite dish, and you need to calculate the precise angle at which to position the receiver. The inverse tangent function helps you determine this angle based on the ratio of the satellite's height and its horizontal distance. By understanding composite functions like tan − 1 ( tan ( x )) , you can ensure that your calculations are accurate and that the receiver is perfectly aligned to capture the satellite signal. This ensures optimal performance and clear communication.
The exact value of the composite function tan − 1 [ tan ( − 19 6 π ) ] is − 19 6 π . This is because the angle − 19 6 π lies within the range of the inverse tangent function, allowing for direct evaluation. Therefore, the correct answer is − 19 6 π .
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