Final answer:
To evaluate tan⁻¹(tan 4), we consider the range of the arctan function and use the periodicity of the tangent function. The answer is 4 - π, which is slightly greater than -π/2 and less than π/2, and within the range of arctan.
Step-by-step explanation:
The question asks to evaluate tan⁻¹(tan 4). This requires understanding the properties of trigonometric functions and their inverses. The inverse tangent function, denoted as tan⁻¹ or arctan, has a range of -π/2 to π/2 (excluding -π/2 and π/2). When you take the tangent of a number and then take the inverse tangent of that result, you essentially undo the tangent function. However, since 4 radians is not within the range of arctan, we need to find an angle within the range that has the same tangent value as 4 radians.
The tangent function has a period of π, meaning that tan(4) is equivalent to tan(4 - nπ), where n is any integer that makes the angle fall within the range of arctan. Since 4 is slightly greater than 4 - π (which is approximately 1.28 radians), this angle falls within the range of arctan, and therefore, tan⁻¹(tan 4) = 4 - π.