Find each exact function value. See Example 3.
tan 5π/3

Verified step by step guidance

1. Recognize that the angle given is \( \frac{5\pi}{3} \), which is in radians. Since \( 2\pi \) radians correspond to a full circle, note that \( \frac{5\pi}{3} \) is between \( \pi \) and \( 2\pi \), specifically in the fourth quadrant.
2. Find the reference angle for \( \frac{5\pi}{3} \) by subtracting it from \( 2\pi \): \( 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3} \). The reference angle is \( \frac{\pi}{3} \).
3. Recall the exact value of \( \tan \) for the reference angle \( \frac{\pi}{3} \). From the unit circle or trigonometric values, \( \tan \frac{\pi}{3} = \sqrt{3} \).
4. Determine the sign of \( \tan \) in the fourth quadrant. Since tangent is sine over cosine, and sine is negative while cosine is positive in the fourth quadrant, tangent is negative there.
5. Combine the reference angle value and the sign to write \( \tan \frac{5\pi}{3} = -\sqrt{3} \).

Key Concepts

Unit Circle and Angle Measurement in Radians

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to arc lengths on this circle. Understanding how to locate angles like 5π/3 on the unit circle helps determine the coordinates and thus the trigonometric function values.

Tangent Function Definition and Properties

The tangent of an angle in the unit circle is defined as the ratio of the y-coordinate to the x-coordinate (sin θ / cos θ). It is periodic with period π and has vertical asymptotes where cosine is zero. Knowing this helps in finding exact values and understanding the function's behavior.

Reference Angles and Quadrant Sign Rules

Reference angles are acute angles formed by the terminal side of the given angle and the x-axis. They simplify finding exact trigonometric values by relating them to known angles. Additionally, the sign of tangent depends on the quadrant where the angle lies, which is crucial for determining the correct value.