Find the indicated function value. If it is undefined, say so. See Example 4. cos 1800°

Verified step by step guidance

1. Convert the angle from degrees to a standard position by finding its coterminal angle. To do this, subtract multiples of 360° from 1800° until the angle is between 0° and 360°.
2. Calculate 1800° - 5 \times 360° = 0°. This means 1800° is coterminal with 0°.
3. Recall that the cosine function is periodic with a period of 360°, so \( \cos(1800°) = \cos(0°) \).
4. Use the unit circle to find \( \cos(0°) \). On the unit circle, the cosine of an angle is the x-coordinate of the corresponding point.
5. The point on the unit circle at 0° is (1, 0), so \( \cos(0°) = 1 \).

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 angle in degrees or radians corresponds to a point on the circle, where the x-coordinate represents the cosine value and the y-coordinate represents the sine value.

Angle Measurement and Coterminal Angles

Angles can be measured in degrees or radians, and angles that differ by full rotations (360° or 2π radians) are called coterminal angles. For example, 1800° can be simplified by subtracting multiples of 360° to find an equivalent angle within the standard range of 0° to 360°. This simplification is crucial for evaluating trigonometric functions.

Cosine Function

The cosine function, denoted as cos(θ), gives the x-coordinate of a point on the unit circle corresponding to an angle θ. It is periodic with a period of 360°, meaning that cos(θ) = cos(θ + 360n) for any integer n. Understanding the properties of the cosine function, including its values at key angles, is essential for solving trigonometric problems.