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

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Recall that the secant function, \( \sec(\theta) \), is the reciprocal of the cosine function, \( \cos(\theta) \). Therefore, \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

Identify the angle given in the problem, which is \( 180^\circ \).

Determine the cosine of \( 180^\circ \). From the unit circle, \( \cos(180^\circ) = -1 \).

Substitute \( \cos(180^\circ) = -1 \) into the reciprocal function: \( \sec(180^\circ) = \frac{1}{-1} \).

Simplify the expression to find \( \sec(180^\circ) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ). This means that wherever the cosine function is zero, the secant function will be undefined, as division by zero is not possible.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it allows for the visualization of the values of trigonometric functions for various angles. The coordinates of points on the unit circle correspond to the cosine and sine values of those angles.

Angle Measurement

Angles can be measured in degrees or radians, with 180° equivalent to π radians. Understanding how to convert between these two systems is crucial in trigonometry. In this case, evaluating sec(180°) requires knowledge of the cosine value at that angle, which is essential for determining the secant value.

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