In Exercises 8–13, find the exact value of each expression. Do not use a calculator.tan 300°

Verified step by step guidance

1. insert step 1> Convert the angle from degrees to radians if necessary, but since we are working with degrees, we can proceed with 300° directly.
2. insert step 2> Recognize that 300° is in the fourth quadrant of the unit circle.
3. insert step 3> Recall that the tangent function is negative in the fourth quadrant.
4. insert step 4> Use the reference angle for 300°, which is 360° - 300° = 60°.
5. insert step 5> Calculate \( \tan(300°) = -\tan(60°) \), and use the known value \( \tan(60°) = \sqrt{3} \).

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 interpretation of the sine, cosine, and tangent functions. Angles measured in degrees or radians correspond to points on the circle, allowing for the determination of exact values for trigonometric functions.

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. For angles greater than 180°, like 300°, the reference angle helps in finding the trigonometric values by relating them to angles in the first quadrant. The reference angle for 300° is 60°, which is crucial for calculating the tangent value.

Tangent Function

The tangent function, defined as the ratio of the sine to the cosine of an angle, is a key trigonometric function. For any angle θ, tan(θ) = sin(θ)/cos(θ). Understanding the signs of sine and cosine in different quadrants is essential for determining the exact value of tangent, especially for angles like 300° that lie in the fourth quadrant.