Find the exact value of each real number y. Do not use a calculator.
y = tan⁻¹ (―√3)

Verified step by step guidance

1. Recognize that \( y = \tan^{-1}(\sqrt{3}) \) implies that \( \tan(y) = \sqrt{3} \).
2. Recall the special angles in trigonometry, specifically those in the unit circle, where the tangent function has known values.
3. Identify the angle \( y \) where \( \tan(y) = \sqrt{3} \). This occurs at \( y = \frac{\pi}{3} \) or \( y = 60^\circ \) in the first quadrant.
4. Consider the range of the inverse tangent function, which is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), to determine the principal value of \( y \).
5. Conclude that the exact value of \( y \) is \( \frac{\pi}{3} \) since it falls within the range of the inverse tangent function.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹, are used to find angles when given a ratio of sides in a right triangle. For example, tan⁻¹(x) gives the angle whose tangent is x. Understanding how these functions work is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Tangent Function

The tangent function relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. It is periodic and has a range of all real numbers. Knowing the values of the tangent function for common angles (like 30°, 45°, and 60°) helps in determining the angles corresponding to specific tangent values, such as -√3.

Quadrants and Angle Values

The unit circle is divided into four quadrants, each corresponding to different signs of the sine, cosine, and tangent functions. For instance, the tangent function is negative in the second and fourth quadrants. Understanding which quadrant an angle lies in is essential for determining the correct angle when solving for inverse trigonometric functions.