In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of0, 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.
cot 15𝜋/2

Verified step by step guidance

1. Identify the angle 15\pi/2 and find its equivalent angle within the unit circle by subtracting multiples of 2\pi.
2. Calculate 15\pi/2 - 6\pi = 3\pi/2 to find the equivalent angle.
3. Locate the angle 3\pi/2 on the unit circle, which corresponds to the point (0, -1).
4. Recall that cotangent is the reciprocal of tangent, so \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}.
5. Substitute the coordinates (0, -1) into the cotangent formula: \cot(3\pi/2) = \frac{0}{-1}.

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 and cosine functions. The coordinates of points on the unit circle correspond to the cosine and sine of angles measured in radians, allowing for the evaluation of trigonometric functions at various angles.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and cotangent, relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, the sine of an angle is the y-coordinate, and the cosine is the x-coordinate of the corresponding point on the circle. The cotangent function, specifically, is defined as the ratio of the cosine to the sine, which can be derived from the coordinates of points on the unit circle.

Periodic Properties

Trigonometric functions exhibit periodic properties, meaning they repeat their values in regular intervals. For example, the sine and cosine functions have a period of 2π, while the cotangent function has a period of π. This property allows for the evaluation of trigonometric functions at angles greater than 2π by reducing them to an equivalent angle within the first cycle, facilitating easier calculations.