Find each exact function value. See Example 3.
tan (-14π/ 3)

Verified step by step guidance

1. First, recognize that the tangent function has a period of \(\pi\), meaning \(\tan(\theta) = \tan(\theta + k\pi)\) for any integer \(k\). This allows us to simplify the angle by adding or subtracting multiples of \(\pi\).
2. Start by simplifying the angle \(-\frac{14\pi}{3}\). Since the period is \(\pi = \frac{3\pi}{3}\), find an integer \(k\) such that \(-\frac{14\pi}{3} + k\pi\) lies within a standard interval, for example between \(-\pi\) and \(\pi\) or between \(0\) and \(2\pi\).
3. Calculate \(k\) by dividing \(-\frac{14}{3}\) by \(1\) (since the period in terms of \(\pi\) is 1), and find the closest integer to add multiples of \(\pi\) to bring the angle into a simpler equivalent angle. For instance, add \(5\pi = \frac{15\pi}{3}\) to \(-\frac{14\pi}{3}\) to get a positive angle.
4. After simplification, express the resulting angle in terms of a known angle on the unit circle, such as \(\frac{\pi}{3}\), \(\frac{2\pi}{3}\), etc., to find the exact value of the tangent function using known tangent values.
5. Finally, use the identity \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) and the known sine and cosine values for the simplified angle to write the exact value of \(\tan(-\frac{14\pi}{3})\).

Key Concepts

Angle Reduction Using Coterminal Angles

Angles differing by full rotations (multiples of 2π) share the same trigonometric values. To simplify tan(-14π/3), add or subtract 2π until the angle lies within a standard interval, such as [0, 2π), making it easier to evaluate.

Tangent Function Properties and Periodicity

The tangent function has a period of π, meaning tan(θ) = tan(θ + π). This property allows further simplification of angles by reducing them modulo π, which helps in finding exact values without a calculator.

Exact Values of Tangent for Special Angles

Certain angles, like π/6, π/4, and π/3, have known exact tangent values (e.g., tan(π/3) = √3). Recognizing the simplified angle as one of these special angles enables direct determination of the exact tangent value.