In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined.cot 𝜋2

Verified step by step guidance

1. Identify the trigonometric function: cotangent, which is the reciprocal of the tangent function.
2. Recognize that the angle given is \( \frac{\pi}{2} \), which is a quadrantal angle.
3. Recall that the tangent of \( \frac{\pi}{2} \) is undefined because it corresponds to a vertical line on the unit circle.
4. Since cotangent is the reciprocal of tangent, if tangent is undefined, cotangent will also be undefined.
5. Conclude that \( \cot \frac{\pi}{2} \) is undefined.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that are multiples of 90 degrees (or π/2 radians) and correspond to the axes on the unit circle. These angles include 0, π/2, π, 3π/2, and 2π. At these angles, the sine and cosine values take on specific values, which are essential for evaluating trigonometric functions.

Cotangent Function

The cotangent function, denoted as cot(θ), is defined as the ratio of the cosine of an angle to the sine of that angle: cot(θ) = cos(θ)/sin(θ). It is important to note that cotangent is undefined when the sine of the angle is zero, which occurs at quadrantal angles like 0, π, and 2π.

Undefined Expressions in Trigonometry

In trigonometry, certain expressions can be undefined due to division by zero. For example, when evaluating cot(π/2), the sine of π/2 is 1, while the cosine is 0, leading to cot(π/2) = 0/1, which is defined. However, cot(0) or cot(π) would be undefined since they involve division by zero.