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

Verified step by step guidance

1. Recall that the secant function is the reciprocal of the cosine function, so \(\sec \theta = \frac{1}{\cos \theta}\).
2. Since the angle given is \(1800^\circ\), reduce it to an equivalent angle between \(0^\circ\) and \(360^\circ\) by subtracting multiples of \(360^\circ\): calculate \(1800^\circ - 5 \times 360^\circ\).
3. Simplify the expression to find the equivalent angle: \(1800^\circ - 1800^\circ = 0^\circ\).
4. Evaluate \(\cos 0^\circ\), knowing that \(\cos 0^\circ = 1\).
5. Find \(\sec 1800^\circ\) by taking the reciprocal of \(\cos 0^\circ\), so \(\sec 1800^\circ = \frac{1}{1}\).

Key Concepts

Understanding the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). It is undefined wherever cos(θ) equals zero, which occurs at odd multiples of 90°. Knowing this helps determine when sec(θ) is defined or undefined.

Angle Coterminality and Reduction

Angles differing by full rotations (360°) share the same trigonometric values. To simplify large angles like 1800°, subtract multiples of 360° to find a coterminal angle between 0° and 360°, making it easier to evaluate the function.

Evaluating Trigonometric Functions at Standard Angles

Trigonometric functions have known values at standard angles such as 0°, 90°, 180°, and 270°. Recognizing these values allows quick evaluation of functions like sec(θ) once the angle is reduced, facilitating identification of defined or undefined values.