Evaluate cos (11π/12) as cos (π/4 + 2π/3).

Verified step by step guidance

1. Recognize that the expression cos(11π/12) can be rewritten using the angle addition formula for cosine: cos(π/4 + 2π/3).
2. Recall the angle addition formula for cosine: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
3. Substitute A = π/4 and B = 2π/3 into the angle addition formula: cos(π/4 + 2π/3) = cos(π/4)cos(2π/3) - sin(π/4)sin(2π/3).
4. Find the values of cos(π/4), cos(2π/3), sin(π/4), and sin(2π/3) using known trigonometric values: cos(π/4) = √2/2, cos(2π/3) = -1/2, sin(π/4) = √2/2, sin(2π/3) = √3/2.
5. Substitute these values into the formula: cos(π/4 + 2π/3) = (√2/2)(-1/2) - (√2/2)(√3/2).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic with a period of 2π, meaning that cos(θ) = cos(θ + 2πn) for any integer n. Understanding the properties of the cosine function is essential for evaluating expressions involving angles.

Angle Addition Formula

The angle addition formula for cosine states that cos(a + b) = cos(a)cos(b) - sin(a)sin(b). This formula allows us to break down the cosine of a sum of angles into simpler components, making it easier to evaluate complex expressions. It is particularly useful when dealing with angles that are not standard, such as π/4 and 2π/3.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric interpretation of trigonometric functions, where the x-coordinate of a point on the circle corresponds to the cosine of the angle formed with the positive x-axis. Familiarity with the unit circle helps in determining the values of trigonometric functions for various angles, including those expressed in radians.