In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of


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


a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
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cot 𝜋/2

Verified step by step guidance

1. Identify the point on the unit circle corresponding to \( \frac{\pi}{2} \). This point is (0, 1).
2. Recall that the cotangent function is defined as \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).
3. For \( \theta = \frac{\pi}{2} \), the coordinates are (0, 1), so \( \cos(\frac{\pi}{2}) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \).
4. Substitute these values into the cotangent formula: \( \cot(\frac{\pi}{2}) = \frac{0}{1} \).
5. Recognize that division by zero is undefined, so \( \cot(\frac{\pi}{2}) \) 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 values of these functions for various angles, allowing for easy calculation of trigonometric values.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and cotangent, relate the angles of a triangle to the lengths of its sides. For example, cotangent is defined as the ratio of the adjacent side to the opposite side in a right triangle. Understanding these functions is crucial for solving problems involving angles and their corresponding values on the unit circle.

Periodic Properties

Trigonometric functions exhibit periodic properties, meaning they repeat their values in regular intervals. For instance, the cotangent function has a period of π, indicating that cot(θ) = cot(θ + nπ) for any integer n. This property allows us to find the values of trigonometric functions at various angles by using known values and adjusting for the periodic nature of these functions.