Find each exact function value. See Example 2.
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: \(\theta = \frac{11\pi}{6}\). This angle is in radians and is located in the fourth quadrant of the unit circle.
3. Find the reference angle for \(\frac{11\pi}{6}\) by subtracting it from \(2\pi\): \(2\pi - \frac{11\pi}{6} = \frac{12\pi}{6} - \frac{11\pi}{6} = \frac{\pi}{6}\).
4. Determine the sine of the reference angle \(\frac{\pi}{6}\), which is \(\sin \frac{\pi}{6} = \frac{1}{2}\). Since \(\frac{11\pi}{6}\) is in the fourth quadrant where sine is negative, \(\sin \frac{11\pi}{6} = -\frac{1}{2}\).
5. Calculate \(\csc \frac{11\pi}{6}\) by taking the reciprocal of \(\sin \frac{11\pi}{6}\): \(\csc \frac{11\pi}{6} = \frac{1}{-\frac{1}{2}}\).

Key Concepts

Understanding the Unit Circle

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. It helps determine the sine, cosine, and other trigonometric values for angles measured in radians. Knowing the coordinates of points on the unit circle corresponding to specific angles allows you to find exact trigonometric function values.

Reciprocal Trigonometric Functions

Cosecant (csc) is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). To find csc for an angle, first find the sine value and then take its reciprocal. This relationship is essential for converting between sine and cosecant values.

Reference Angles and Quadrants

Reference angles help find trigonometric values for angles not on the first quadrant by relating them to acute angles. The sign of the function depends on the quadrant where the angle lies. For 11π/6, understanding its quadrant and reference angle is key to determining the correct sign and exact value.