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

검증된 단계별 안내

Recognize that the cosine function is periodic with a period of 360°, meaning that $\cos(\theta) = \cos(\theta + 360°k)$ for any integer $k$.

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°$.

Perform the calculation: $1800° - 360° \times 5 = 1800° - 1800° = 0°$. So, $\cos 1800° = \cos 0°$.

Recall the value of $\cos 0°$, which is a fundamental trigonometric value.

Conclude that $\cos 1800°$ is equal to $\cos 0°$ and state the corresponding cosine value.

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주요 개념

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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.