Find the indicated function value. If it is undefined, say so. See Example 4. sec 180°

Verified step by step guidance

1. Recall the definition of the secant function: \(\sec \theta = \frac{1}{\cos \theta}\).
2. Identify the angle given: \(180^\circ\).
3. Find the cosine of \(180^\circ\): \(\cos 180^\circ\).
4. Evaluate \(\cos 180^\circ\) using the unit circle or known values. (Note: \(\cos 180^\circ = -1\).)
5. Calculate \(\sec 180^\circ\) by taking the reciprocal of \(\cos 180^\circ\), i.e., \(\sec 180^\circ = \frac{1}{\cos 180^\circ}\).

Key Concepts

Definition of the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function: sec(θ) = 1/cos(θ). It is defined wherever cosine is not zero. Understanding this reciprocal relationship is essential to evaluate secant values for given angles.

Cosine of Special Angles

Cosine values for special angles like 0°, 90°, 180°, and 270° are fundamental. For 180°, cos(180°) = -1. Knowing these values helps directly compute sec(180°) by taking the reciprocal of cosine at that angle.

Domain and Undefined Values of Trigonometric Functions

Trigonometric functions can be undefined when their denominators are zero. Since sec(θ) = 1/cos(θ), sec(θ) is undefined where cos(θ) = 0. Recognizing these points prevents errors in evaluation and helps identify when a function value does not exist.