Find each exact function value. See Example 2.
tan 5π/6

Verified step by step guidance

1. Recall that the tangent function is defined as the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
2. Identify the reference angle for \(\frac{5\pi}{6}\). Since \(\frac{5\pi}{6} = \pi - \frac{\pi}{6}\), the reference angle is \(\frac{\pi}{6}\).
3. Determine the signs of sine and cosine in the second quadrant (where \(\frac{5\pi}{6}\) lies). In the second quadrant, sine is positive and cosine is negative.
4. Use the known exact values for sine and cosine at the reference angle \(\frac{\pi}{6}\): \(\sin \frac{\pi}{6} = \frac{1}{2}\) and \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\).
5. Apply the signs and calculate \(\tan \frac{5\pi}{6} = \frac{\sin \frac{5\pi}{6}}{\cos \frac{5\pi}{6}} = \frac{\sin \frac{\pi}{6}}{-\cos \frac{\pi}{6}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}\). Simplify this expression to find the exact value.

Key Concepts

Unit Circle and Reference Angles

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Reference angles help find the function values by relating any angle to an acute angle in the first quadrant, simplifying calculations.

Tangent Function Definition

Tangent of an angle in the unit circle is the ratio of the y-coordinate to the x-coordinate (sin θ / cos θ). It can also be understood as the slope of the line formed by the angle, and its sign depends on the quadrant where the angle lies.

Evaluating Trigonometric Functions at Special Angles

Special angles like π/6, π/4, and π/3 have known exact sine, cosine, and tangent values. Using these known values and the angle’s quadrant, one can determine the exact value of trigonometric functions for angles like 5π/6.