Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.


cos (2π/3)

Verified step by step guidance

1. Recall the unit circle and the definition of cosine: Cosine of an angle \( \theta \) is the x-coordinate of the point on the unit circle corresponding to \( \theta \).
2. Convert the angle \( \frac{2\pi}{3} \) to degrees if necessary: \( \frac{2\pi}{3} \) radians is equivalent to 120 degrees.
3. Identify the reference angle: The reference angle for \( 120^\circ \) is \( 180^\circ - 120^\circ = 60^\circ \).
4. Determine the sign of the cosine in the second quadrant: In the second quadrant, cosine values are negative.
5. Use the reference angle to find the cosine value: The cosine of \( 60^\circ \) is \( \frac{1}{2} \), so \( \cos(120^\circ) = -\frac{1}{2} \).

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 allows for the definition of sine, cosine, and tangent functions based on the coordinates of points on the circle. For any angle θ, the x-coordinate corresponds to cos(θ) and the y-coordinate corresponds to sin(θ).

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help in evaluating trigonometric functions for angles greater than 90 degrees or less than 0 degrees by relating them to their corresponding acute angles. For example, the reference angle for 2π/3 is π/3, which is used to find the cosine value.

Cosine Function

The cosine function, denoted as cos(θ), represents the x-coordinate of a point on the unit circle corresponding to the angle θ. It is periodic with a period of 2π and has specific values for common angles. For instance, cos(π/3) equals 1/2, and since 2π/3 is in the second quadrant, where cosine is negative, cos(2π/3) equals -1/2.