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

Verified step by step guidance

1. Recall that the cosecant function is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\).
2. Identify the angle given: \(\pi\) radians, which is a quadrantal angle located on the negative x-axis of the unit circle.
3. Evaluate \(\sin \pi\). Since \(\sin \pi = 0\), substitute this value into the reciprocal expression for cosecant.
4. Since \(\csc \pi = \frac{1}{\sin \pi} = \frac{1}{0}\), recognize that division by zero is undefined.
5. Conclude that \(\csc \pi\) is undefined because the sine of \(\pi\) is zero, making the reciprocal undefined.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis in the unit circle, typically multiples of π/2 (e.g., 0, π/2, π, 3π/2). These angles have special sine and cosine values, often 0, ±1, which affect the evaluation of trigonometric functions.

Cosecant Function (csc)

The cosecant function is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). It is undefined wherever sin(θ) = 0, which commonly occurs at quadrantal angles like 0, π, and 2π.

Evaluating Trigonometric Functions at Specific Angles

To evaluate trigonometric functions at specific angles, substitute the angle into the function and use known sine and cosine values from the unit circle. For quadrantal angles, check if the function is defined or undefined due to division by zero.