Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problema 21

Capitolo 4, Problema 21

Find each exact function value. See Example 2.
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: \(\theta = \frac{11\pi}{6}\). This angle is in radians and is located in the fourth quadrant of the unit circle.

Find the reference angle for \(\frac{11\pi}{6}\) by subtracting it from \(2\pi\): \(2\pi - \frac{11\pi}{6} = \frac{12\pi}{6} - \frac{11\pi}{6} = \frac{\pi}{6}\).

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, \(\sin \frac{11\pi}{6} = -\frac{1}{2}\).

Calculate \(\csc \frac{11\pi}{6}\) by taking the reciprocal of \(\sin \frac{11\pi}{6}\): \(\csc \frac{11\pi}{6} = \frac{1}{-\frac{1}{2}}\).

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 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.

Video consigliato:

Percorso guidato

06:11

Introduction to the Unit Circle

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.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

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.

Video consigliato:

Percorso guidato

5:31

Reference Angles on the Unit Circle

Pratica correlata

Domanda del libro di testo

Concept Check If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

608

views

Domanda del libro di testo

Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Assume the radius of Earth is 6400 km. See Example 2. Panama City, Panama, 9° N, and Pittsburgh, Pennsylvania, 40° N

590

views

Domanda del libro di testo

Use the formula v = r ω to find the value of the missing variable.

v = 9 m per sec , r = 5 m

905

views

Domanda del libro di testo

Use the formula v = r ω to find the value of the missing variable.

r = 12 m , ω = 2π/3 radians per sec

786

views