Find the exact value of each expression. (Do not use a calculator.)
cos (-7π/12)

Verified step by step guidance

1. \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
2. \(-\frac{7\pi}{12} = -\frac{3\pi}{4} + \frac{\pi}{3}\)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and coordinates in a two-dimensional plane. It is a circle with a radius of one centered at the origin of a coordinate system. Angles measured in radians correspond to points on the circle, where the x-coordinate represents the cosine value and the y-coordinate represents the sine value of the angle.

Cosine Function

The cosine function is one of the primary trigonometric functions, defined as the ratio of the adjacent side to the hypotenuse in a right triangle. In the context of the unit circle, it gives the x-coordinate of a point on the circle corresponding to a given angle. Understanding the properties of the cosine function, including its periodicity and symmetry, is essential for evaluating expressions like cos(-7π/12).

Angle Reference and Quadrants

When dealing with angles in trigonometry, it is important to understand reference angles and the quadrants of the unit circle. The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. The sign of the cosine value depends on the quadrant in which the angle lies, which can be determined by the angle's measure. For negative angles, the angle is measured clockwise from the positive x-axis.