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

Verified step by step guidance

1. Step 1: Understand the periodicity of the tangent function. The tangent function, \( \tan(\theta) \), has a period of \( \pi \). This means that \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \).
2. Step 2: Simplify the angle \( -\frac{14\pi}{3} \) by adding or subtracting multiples of \( \pi \) to find an equivalent angle within the range \( [0, \pi) \).
3. Step 3: Calculate the equivalent angle by adding \( 5\pi \) (which is \( 15\pi/3 \)) to \( -\frac{14\pi}{3} \) to bring it within the desired range.
4. Step 4: Simplify the resulting angle to find \( \theta \) such that \( \tan(-\frac{14\pi}{3}) = \tan(\theta) \).
5. Step 5: Evaluate \( \tan(\theta) \) using known values or trigonometric identities to find the exact function value.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. Angles measured in radians correspond to points on the circle, allowing for the determination of exact function values for various angles.

Periodic Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For example, the tangent function has a period of π, which means that tan(θ) = tan(θ + nπ) for any integer n. This property is crucial when evaluating angles outside the standard range, as it allows us to find equivalent angles that yield the same function value.

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is used to simplify the calculation of trigonometric function values for angles that are not standard. By finding the reference angle, one can determine the function value in the correct quadrant, which is essential for angles like -14π/3.