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.
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cot 𝜋/2

Verified step by step guidance

1. Step 1: Understand the problem context. The problem involves the unit circle divided into eight equal arcs, each corresponding to specific t-values (angles) measured in radians. The goal is to find the value of the trigonometric function cotangent at \( \frac{\pi}{2} \).
2. Step 2: Recall the definition of cotangent in terms of coordinates on the unit circle. For an angle \( t \), the coordinates on the unit circle are \( (x, y) = (\cos t, \sin t) \). The cotangent function is defined as \( \cot t = \frac{\cos t}{\sin t} = \frac{x}{y} \).
3. Step 3: Identify the coordinates at \( t = \frac{\pi}{2} \) on the unit circle. At this angle, the point on the unit circle is \( (x, y) = (\cos \frac{\pi}{2}, \sin \frac{\pi}{2}) \). Use these coordinates to express \( \cot \frac{\pi}{2} = \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}} \).
4. Step 4: Use the periodic properties of the cotangent function to find its value at other angles if needed. Cotangent has a period of \( \pi \), so \( \cot(t + \pi) = \cot t \). This property helps to find cotangent values at angles beyond the first revolution by relating them back to known values.
5. Step 5: Apply the above steps to evaluate \( \cot \frac{\pi}{2} \) and then use periodicity if asked to find cotangent at other indicated real numbers.

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to points on the circle, where the x-coordinate is cos(θ) and the y-coordinate is sin(θ). Understanding how angles divide the circle into arcs helps locate points and evaluate trigonometric functions.

Definition of Cotangent Function

Cotangent (cot θ) is the ratio of the cosine to the sine of an angle, cot θ = cos θ / sin θ. It is undefined where sin θ = 0, such as at θ = 0 or π. Knowing this ratio and its domain restrictions is essential for evaluating cotangent values at specific angles on the unit circle.

Periodic Properties of Trigonometric Functions

Trigonometric functions repeat their values in regular intervals called periods. For cotangent, the period is π, meaning cot(θ + π) = cot θ. This property allows finding function values at angles beyond the initial interval by relating them back to known values within one period.