In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.6 3 2 3 6 6 3 2 3 6Use 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.
In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.tan 3𝜋/2

Verified step by step guidance

1. Identify the angle \( \frac{3\pi}{2} \) on the unit circle.
2. Locate the coordinates corresponding to \( \frac{3\pi}{2} \) on the unit circle, which are \((0, -1)\).
3. Recall that the tangent function \( \tan(\theta) \) is defined as \( \frac{y}{x} \) where \((x, y)\) are the coordinates on the unit circle.
4. Substitute the coordinates \((0, -1)\) into the tangent function: \( \tan\left(\frac{3\pi}{2}\right) = \frac{-1}{0} \).
5. Recognize that division by zero is undefined, so \( \tan\left(\frac{3\pi}{2}\right) \) is undefined.

Key Concepts

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, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. On the unit circle, these functions can be defined as follows: sine is the y-coordinate, cosine is the x-coordinate, and tangent is the ratio of sine to cosine. Understanding these functions is crucial for solving problems involving angles and their corresponding values.

Undefined Values

Certain trigonometric functions can be undefined for specific angles. For example, the tangent function is undefined when the cosine value is zero, which occurs at angles like 𝜋/2 and 3𝜋/2. Recognizing when a function is undefined is essential for accurately interpreting and solving trigonometric equations, particularly when using the unit circle.