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

Verified step by step guidance

1. Recognize that the cosine function is periodic with a period of 360°, meaning that \(\cos(\theta) = \cos(\theta + 360°k)\) for any integer \(k\).
2. To simplify \(\cos 1800°\), reduce the angle by subtracting multiples of 360° until the angle lies within the standard range of \$0°\( to \)360°\(. Calculate \(1800° - 360° \times k\) where \)k\( is chosen so the result is between \)0°\( and \)360°$.
3. Perform the calculation: \(1800° - 360° \times 5 = 1800° - 1800° = 0°\). So, \(\cos 1800° = \cos 0°\).
4. Recall the value of \(\cos 0°\), which is a fundamental trigonometric value.
5. Conclude that \(\cos 1800°\) is equal to \(\cos 0°\) and state the corresponding cosine value.

Key Concepts

Angle Measurement and Coterminal Angles

Angles can be measured in degrees and can exceed 360°, representing multiple rotations. Coterminal angles differ by full rotations of 360° and share the same trigonometric values. To find the value of a function at a large angle, reduce it by subtracting multiples of 360° to find an equivalent angle within 0° to 360°.

Cosine Function Properties

The cosine function relates an angle to the x-coordinate of a point on the unit circle. It is periodic with a period of 360°, meaning cos(θ) = cos(θ + 360°k) for any integer k. Cosine values range between -1 and 1 and are defined for all real angles.

Evaluating Trigonometric Functions at Specific Angles

To evaluate cos 1800°, first find the coterminal angle by subtracting multiples of 360°. For example, 1800° - 5×360° = 0°, so cos 1800° = cos 0° = 1. This method simplifies calculations and helps determine if the function value is defined or undefined.