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.
tan 0

Verified step by step guidance

1. Identify the angle t = 0 on the unit circle.
2. Locate the coordinates corresponding to t = 0, which are (1, 0).
3. Recall that the tangent function is defined as \( \tan(t) = \frac{y}{x} \), where (x, y) are the coordinates on the unit circle.
4. Substitute the coordinates (1, 0) into the tangent function: \( \tan(0) = \frac{0}{1} \).
5. Simplify the expression to find the value of \( \tan(0) \).

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 these 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, the sine of an angle is the y-coordinate, while the cosine is the x-coordinate of the corresponding point. The tangent function, defined as the ratio of sine to cosine, can be interpreted as the slope of the line formed by the angle's terminal side, which is crucial for solving problems involving angles and their trigonometric values.

Undefined Values

Certain trigonometric functions can be undefined for specific angles. For example, the tangent function is undefined when the cosine of the angle is zero, as this leads to division by zero. Understanding where these undefined values occur on the unit circle is essential for accurately evaluating trigonometric expressions and ensuring correct interpretations of angles and their corresponding functions.