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

Guida verificata passo dopo passo

Recall that the cosecant function is the reciprocal of the sine function, so \(\csc(\theta) = \frac{1}{\sin(\theta)}\).

Identify the angle given: \(-\frac{11\pi}{6}\). Since this is a negative angle, find its positive coterminal angle by adding \(2\pi\): \(-\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6}\).

Evaluate \(\sin\left(\frac{\pi}{6}\right)\) using known special angles. Recall that \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\).

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 \(\frac{1}{2}\).

Finally, find \(\csc\left(-\frac{11\pi}{6}\right)\) by taking the reciprocal of the sine value: \(\csc\left(-\frac{11\pi}{6}\right) = \frac{1}{\sin\left(-\frac{11\pi}{6}\right)} = \frac{1}{\frac{1}{2}}\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.

Video consigliato:

Percorso guidato

6:22

Graphs of Secant and Cosecant Functions

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.

Video consigliato:

Percorso guidato

7:28

Evaluate Composite Functions - Values Not on Unit Circle

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.

Video consigliato:

Percorso guidato