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


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


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. Step 1: Recognize that the problem asks for \( \cot(15\pi/2) \). The cotangent function is defined as \( \cot t = \frac{\cos t}{\sin t} \).
2. Step 2: Use the periodicity of the cotangent function. Since cotangent has a period of \( \pi \), reduce \( 15\pi/2 \) by subtracting multiples of \( \pi \) to find an equivalent angle within the first cycle. Calculate \( 15\pi/2 - 7\pi = (15/2 - 7)\pi = (15/2 - 14/2)\pi = \pi/2 \). So, \( \cot(15\pi/2) = \cot(\pi/2) \).
3. Step 3: Identify the coordinates on the unit circle corresponding to \( t = \pi/2 \). From the figure, the coordinates are \( (0, 1) \), where \( x = \cos(\pi/2) = 0 \) and \( y = \sin(\pi/2) = 1 \).
4. Step 4: Calculate \( \cot(\pi/2) = \frac{\cos(\pi/2)}{\sin(\pi/2)} = \frac{0}{1} \).
5. Step 5: Interpret the result from step 4 to understand the behavior of cotangent at \( \pi/2 \) and relate it back to the original angle.

Key Concepts

Unit Circle and Coordinates

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t, measured in radians, and has coordinates (x, y) = (cos t, sin t). These coordinates are essential for evaluating trigonometric functions at specific angles.

Cotangent Function

The cotangent of an angle t, cot(t), is defined as the ratio of the cosine to the sine of that angle: cot(t) = cos(t)/sin(t). Using the coordinates from the unit circle, cot(t) can be found by dividing the x-coordinate by the y-coordinate of the corresponding point.

Periodicity of Trigonometric Functions

Trigonometric functions like sine, cosine, and cotangent are periodic, meaning their values repeat at regular intervals. For cotangent, the period is π, so cot(t + kπ) = cot(t) for any integer k. This property allows simplification of angles outside the standard interval by reducing them modulo the period.