Find exact values of the six trigonometric functions for each angle. Do not use a calculator. Rationalize denominators when applicable. 120°

Verified step by step guidance

1. Recognize that 120° is in the second quadrant, where sine is positive and cosine is negative.
2. Express 120° as 180° - 60°, so use the reference angle 60° to find the trigonometric values.
3. Use the known exact values for 60°: \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), and \(\tan 60^\circ = \sqrt{3}\).
4. Apply the signs for the second quadrant: \(\sin 120^\circ = \sin 60^\circ\), \(\cos 120^\circ = -\cos 60^\circ\), and \(\tan 120^\circ = -\tan 60^\circ\).
5. Find the reciprocal functions using the definitions: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\), then rationalize denominators if needed.

Key Concepts

Reference Angles and Quadrants

To find trigonometric values for angles like 120°, identify the reference angle by subtracting from 180°, giving 60°. Recognize that 120° lies in the second quadrant, where sine is positive and cosine and tangent are negative. This helps determine the sign of each function.

Exact Values of Trigonometric Functions for Special Angles

Certain angles such as 30°, 45°, and 60° have known exact trigonometric values involving square roots and fractions. For 60°, sine is √3/2, cosine is 1/2, and tangent is √3. Using these exact values avoids approximations and calculator use.

Rationalizing Denominators

When trigonometric values have denominators with square roots, rationalize by multiplying numerator and denominator by the root to eliminate it. For example, convert 1/√3 to √3/3. This is a standard practice to present answers in simplified, exact form.