In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. cos 225°

Verified step by step guidance

1. Identify the quadrant in which the angle 225° lies. Since 225° is between 180° and 270°, it is in the third quadrant.
2. Find the reference angle for 225°. The reference angle is the acute angle formed with the x-axis, calculated as \$225° - 180°$.
3. Recall the cosine values for the reference angle. Since the reference angle is \$45°$, use the known exact value \(\cos 45° = \frac{\sqrt{2}}{2}\).
4. Determine the sign of cosine in the third quadrant. Cosine is negative in the third quadrant, so the value of \(\cos 225°\) will be negative.
5. Combine the sign and the reference angle cosine value to express \(\cos 225°\) as \(-\frac{\sqrt{2}}{2}\).

Key Concepts

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating angles in different quadrants to their acute counterparts, allowing the use of known values for sine, cosine, and tangent.

Unit Circle and Quadrants

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Understanding which quadrant an angle lies in determines the sign (positive or negative) of the trigonometric values based on the coordinate signs in that quadrant.

Exact Values of Common Angles

Certain angles like 30°, 45°, 60°, and their multiples have known exact trigonometric values expressed in simplified radical form. Using these exact values avoids approximation and is essential when calculating trigonometric functions without a calculator.