Find the exact value, if any, of the following composite function. Do not use a calculator.
[tex]\tan ^{-1}\left[\tan \left(-\frac{3 \pi}{13}\right)\right][/tex]
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. [tex]\tan ^{-1}\left[\tan \left(-\frac{3 \pi}{13}\right)\right]=[/tex]
(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 (1)
The problem asks to find the exact value of tan − 1 ( tan ( − 13 3 π )) .
The range of tan − 1 ( x ) is ( − 2 π , 2 π ) .
Since − 13 3 π is within the range ( − 2 π , 2 π ) , the expression simplifies to − 13 3 π .
Therefore, the final answer is − 13 3 π .
Explanation
Understanding the Problem We are asked to find the exact value of the composite function tan − 1 [ tan ( − 13 3 π ) ] . The inverse tangent function, denoted as tan − 1 ( x ) or arctan ( x ) , has a range of ( − 2 π , 2 π ) . This means that the output of the tan − 1 function must lie within this interval.
Checking the Interval To evaluate the composite function, we need to check if the argument of the outer function, tan − 1 , is within its domain. In this case, we need to determine if − 13 3 π is within the interval ( − 2 π , 2 π ) .
Evaluating the Composite Function Since − 2 π < − 13 3 π < 2 π , the value − 13 3 π is within the range of the inverse tangent function. Therefore, the composite function is defined, and we have tan − 1 [ tan ( − 13 3 π ) ] = − 13 3 π .
Final Answer Thus, the exact value of the composite function is − 13 3 π .
Examples
Imagine you're designing a satellite dish aiming to capture signals from a specific angle. The arctangent function helps you calculate the precise angle needed for the dish's orientation. If you know the ratio of the height to the distance of the signal source, you can use arctangent to find the angle. This ensures the dish is perfectly aligned to receive the strongest signal, optimizing performance. This is also applicable in fields like astronomy, where precise angular measurements are crucial for observing celestial objects.
