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

Verified step by step guidance

1. Recall that the tangent function has a period of 180°, so \( \tan(300^\circ) = \tan(300^\circ - 180^\circ) = \tan(120^\circ) \).
2. Identify the reference angle for 120°. Since 120° is in the second quadrant, the reference angle is \( 180^\circ - 120^\circ = 60^\circ \).
3. Determine the sign of tangent in the second quadrant. Tangent is negative in the second quadrant because sine is positive and cosine is negative, and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
4. Use the exact value of \( \tan 60^\circ \), which is \( \sqrt{3} \).
5. Combine the sign and the reference angle value to write \( \tan 300^\circ = -\sqrt{3} \).

Key Concepts

Reference Angles and Quadrants

Understanding reference angles helps simplify trigonometric values by relating them to acute angles. The angle 300° lies in the fourth quadrant, where tangent values are negative. Identifying the quadrant determines the sign of the trigonometric function.

Tangent Function Definition

The tangent of an angle in standard position is the ratio of the sine to the cosine of that angle (tan θ = sin θ / cos θ). Knowing this relationship allows calculation of tangent values using known sine and cosine values of reference angles.

Exact Values of Special Angles

Certain angles like 30°, 45°, and 60° have well-known exact sine, cosine, and tangent values. Since 300° corresponds to 360° - 60°, using the exact values for 60° helps find the exact tangent of 300° without a calculator.