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

Verified step by step guidance

1. Convert the angle from radians to degrees. Since \( \pi \) radians is equal to 180 degrees, multiply \( \frac{5\pi}{3} \) by \( \frac{180}{\pi} \) to find the equivalent angle in degrees.
2. Simplify the angle in degrees to find its coterminal angle between 0 and 360 degrees by subtracting 360 degrees if necessary.
3. Identify the reference angle. The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis.
4. Determine the sign of the tangent function in the quadrant where the angle lies. Recall that tangent is positive in the first and third quadrants and negative in the second and fourth quadrants.
5. Use the reference angle to find the exact value of the tangent function. Use known values of tangent for common angles such as 30°, 45°, and 60° to determine the exact 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 allows for the definition of sine, cosine, and tangent functions based on the coordinates of points on the circle. For any angle θ, the coordinates (cos(θ), sin(θ)) represent the cosine and sine values, respectively, which are essential for calculating trigonometric functions like tangent.

Tangent Function

The tangent function, denoted as tan(θ), is defined as the ratio of the sine and cosine of an angle: tan(θ) = sin(θ) / cos(θ). It represents the slope of the line formed by the angle in the unit circle. Understanding how to compute tangent values using the unit circle is crucial for finding exact function values for specific angles, such as 5π/3.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help simplify the calculation of trigonometric functions for angles greater than 90 degrees. For example, to find tan(5π/3), one can determine its reference angle, which is 2π/3, and use the properties of the tangent function in the appropriate quadrant to find the exact value.