Find each exact function value.
csc ( ―11π/6)

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: \(-\frac{11\pi}{6}\). Since this is a negative angle, find its positive coterminal angle by adding \(2\pi\): \(-\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6}\).
3. Evaluate \(\sin\left(\frac{\pi}{6}\right)\) using known special angles. Recall that \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\).
4. Since \(\sin\left(-\frac{11\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right)\) (because sine is an odd function and the angle is coterminal), the value is \(\frac{1}{2}\).
5. Finally, find \(\csc\left(-\frac{11\pi}{6}\right)\) by taking the reciprocal of the sine value: \(\csc\left(-\frac{11\pi}{6}\right) = \frac{1}{\sin\left(-\frac{11\pi}{6}\right)} = \frac{1}{\frac{1}{2}}\).

Key Concepts

Understanding the Cosecant Function

The cosecant function, csc(θ), is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). To find csc(θ), you first determine sin(θ) and then take its reciprocal. This relationship is fundamental when evaluating cosecant values exactly.

Evaluating Trigonometric Functions at Negative Angles

Negative angles in trigonometry represent clockwise rotation from the positive x-axis. The sine function is odd, meaning sin(-θ) = -sin(θ). This property helps convert negative angle values into positive ones for easier evaluation using known reference angles.

Reference Angles and Unit Circle Values

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. Using the unit circle, you can find exact sine values for common angles like π/6, π/4, and π/3. Recognizing the reference angle for -11π/6 allows precise calculation of sine and thus cosecant.