Find the exact value of each expression. (Do not use a calculator.)
cos π/12

Verified step by step guidance

1. Recognize that \( \frac{\pi}{12} \) is an angle that can be expressed as a difference of two special angles: \( \frac{\pi}{3} - \frac{\pi}{4} \).
2. Use the cosine difference identity: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
3. Substitute \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \) into the identity: \( \cos(\frac{\pi}{3} - \frac{\pi}{4}) = \cos(\frac{\pi}{3})\cos(\frac{\pi}{4}) + \sin(\frac{\pi}{3})\sin(\frac{\pi}{4}) \).
4. Recall the exact values: \( \cos(\frac{\pi}{3}) = \frac{1}{2} \), \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \), and \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
5. Substitute these values into the expression and simplify: \( \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \).

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 and cosine functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured from the positive x-axis, allowing for the determination of exact values for trigonometric functions.

Angle Sum and Difference Identities

Angle sum and difference identities are formulas that express the sine and cosine of the sum or difference of two angles in terms of the sine and cosine of those angles. For example, cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b). These identities are particularly useful for finding exact values of trigonometric functions for angles that are not standard, such as π/12, by expressing them as sums or differences of known angles.

Special Angles

Special angles in trigonometry refer to angles that have known sine and cosine values, typically 0, π/6, π/4, π/3, and π/2. Understanding these angles allows for easier calculations and derivations of trigonometric values. For instance, π/12 can be expressed as π/4 - π/6, enabling the use of angle difference identities to find its cosine value without a calculator.