In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.
6 3 2 3 6 6 3 2 3 6

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.
sec 3𝜋/2

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Identify the angle \( \frac{3\pi}{2} \) on the unit circle. This angle corresponds to 270 degrees, which is located on the negative y-axis.

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

Determine the coordinates of the point on the unit circle at \( \frac{3\pi}{2} \). The coordinates are (0, -1).

Find the cosine of \( \frac{3\pi}{2} \) using the x-coordinate of the point, which is 0. Thus, \( \cos(\frac{3\pi}{2}) = 0 \).

Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), and \( \cos(\frac{3\pi}{2}) = 0 \), the expression \( \sec(\frac{3\pi}{2}) \) is undefined because division by zero is undefined.

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Key Concepts

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

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, allowing for easy calculation of trigonometric functions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and secant, relate the angles of a triangle to the lengths of its sides. For example, the secant function is defined as the reciprocal of the cosine function. Understanding these functions is crucial for evaluating expressions like sec(3π/2), which requires knowledge of the angle's position on the unit circle and the corresponding coordinates.

Undefined Expressions

Certain trigonometric functions can be undefined for specific angles. For instance, secant is undefined when the cosine of the angle is zero, which occurs at odd multiples of π/2. Recognizing when a trigonometric function is undefined is essential for accurately solving problems and interpreting results in trigonometry.

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