In Exercises 1–26, find the exact value of each expression. _ tan⁻¹ (−√3)

Verified step by step guidance

1. Recognize that the expression is asking for the inverse tangent (arctangent) of \(-\sqrt{3}\), which means we want to find an angle \(\theta\) such that \(\tan(\theta) = -\sqrt{3}\).
2. Recall the basic angles where tangent values are known: \(\tan(\frac{\pi}{3}) = \sqrt{3}\) and \(\tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}\). Since the value is \(-\sqrt{3}\), the angle must correspond to the negative of \(\tan(\frac{\pi}{3})\).
3. Determine the principal value range of the inverse tangent function, which is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), meaning the angle must lie in the first or fourth quadrant.
4. Find the angle in the fourth quadrant where tangent is negative and equals \(-\sqrt{3}\). This angle is the negative of \(\frac{\pi}{3}\), so \(\theta = -\frac{\pi}{3}\).
5. Express the final answer as \(\tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3}\), which is the exact value within the principal range.

Key Concepts

Inverse Tangent Function (tan⁻¹ or arctan)

The inverse tangent function, denoted as tan⁻¹ or arctan, returns the angle whose tangent is a given number. It is used to find an angle when the ratio of the opposite side to the adjacent side in a right triangle is known. The output angle is typically in the range (-π/2, π/2) or (-90°, 90°).

Exact Values of Tangent for Special Angles

Certain angles have well-known tangent values, such as π/6, π/4, and π/3. For example, tan(π/3) = √3, so tan⁻¹(√3) = π/3. Recognizing these special values helps in finding exact angle measures without a calculator.

Handling Negative Arguments in Inverse Trigonometric Functions

When the input to an inverse trig function is negative, the resulting angle lies in the corresponding negative range of the function's principal values. For tan⁻¹(−√3), the angle is negative and corresponds to the angle whose tangent is √3 but reflected across the x-axis, typically -π/3.