Select the correct choice and, if necessary, fill in the answer box within your choice.
A.$\tan ^{-1}\left[\tan \left(-\frac{2 \pi}{3}\right)\right]=$ $\square$ (Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers
B. It is not defined.

The problem requires evaluating tan − 1 ( tan ( − 3 2 π ​ )) .
The range of tan − 1 ( x ) is ( − 2 π ​ , 2 π ​ ) .
Use the property tan ( x ) = tan ( x + nπ ) to find an equivalent angle in the desired range.
Adding π to − 3 2 π ​ gives 3 π ​ , which is in the range of tan − 1 ( x ) .
Therefore, the final answer is 3 π ​ ​ .

Explanation

Understanding the Problem We are asked to evaluate the expression tan − 1 ( tan ( − 3 2 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 be an angle between − 2 π ​ and 2 π ​ .

Using the Periodicity of Tangent The tangent function has a period of π , which means that tan ( x ) = tan ( x + nπ ) for any integer n . We want to find an angle θ in the interval ( − 2 π ​ , 2 π ​ ) such that tan ( θ ) = tan ( − 3 2 π ​ ) .

Finding an Equivalent Angle To find such an angle, we can add or subtract multiples of π from − 3 2 π ​ until we obtain an angle in the interval ( − 2 π ​ , 2 π ​ ) . Let's add π to − 3 2 π ​ : − 3 2 π ​ + π = 1 − 3 2 π ​ + 3 3 π ​ ​ = 3 π ​ .

Evaluating the Expression Since 3 π ​ is in the interval ( − 2 π ​ , 2 π ​ ) , we have found the angle we are looking for. Therefore, tan − 1 ( tan ( − 3 2 π ​ )) = 3 π ​ .

Final Answer Thus, the simplified answer is 3 π ​ .


Examples
Imagine you're designing a robotic arm that needs to reach a specific point. The angles of the joints can be described using trigonometric functions. The inverse tangent function helps you determine the correct angle for a joint, given the desired position of the arm. By understanding how to simplify expressions involving inverse trigonometric functions, you can ensure the robotic arm moves accurately and efficiently. This is crucial in manufacturing, surgery, and other applications where precision is key.

Answered by GinnyAnswer | 2025-07-07

The expression tan − 1 [ tan ( − 3 2 π ​ ) ] simplifies to 3 π ​ since the tangent function is periodic and the angle lies in the appropriate range for the inverse tangent. The calculation shows that we can convert the original angle to a coterminal one that is more manageable. Thus, the final answer is 3 π ​ .
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Answered by Anonymous | 2025-07-22