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 o
B. It is not defined.
Answer (1)
Recognize that the range of tan − 1 ( x ) is ( − 2 π , 2 π ) .
Use the periodicity of the tangent function to find an angle in the range of the inverse tangent.
Simplify − 3 2 π by adding π to get 3 π .
Evaluate the expression: tan − 1 ( tan ( − 3 2 π )) = 3 π , so the final answer is 3 π .
Explanation
Understanding the Problem We are asked to evaluate the expression tan − 1 ( tan ( − 3 2 π )) or state that it is not defined. The inverse tangent function, denoted as tan − 1 ( x ) or arctan ( x ) , returns an angle whose tangent is x . The range of the inverse tangent function is ( − 2 π , 2 π ) , which means the output of tan − 1 ( x ) must be 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 Since the tangent function has a period of π , we can add or subtract multiples of π from − 3 2 π to find an equivalent angle within the range of the inverse tangent function. Adding π to − 3 2 π , we get: − 3 2 π + π = 1 − 3 2 π + 3 3 π = 3 π Since 3 π is in the interval ( − 2 π , 2 π ) , we can use this angle.
Evaluating the Expression Now we can evaluate the expression: tan − 1 ( tan ( − 3 2 π )) = tan − 1 ( tan ( 3 π )) = 3 π Therefore, the answer is 3 π .
Final Answer The expression tan − 1 ( tan ( − 3 2 π )) simplifies to 3 π .
Examples
Imagine you're piloting an aircraft and need to determine your heading relative to a landmark. The arctangent function helps convert the ratio of your displacement components (eastward and northward) into an angle, providing your direction. Similarly, in robotics, calculating joint angles for arm movements often involves inverse trigonometric functions, ensuring precise positioning and control.
