Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problema 1.3.69

Capitolo 1, Problema 1.3.69

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. csc(7𝜋/6)

Guida verificata passo dopo passo

Identify the given angle: \(7\pi/6\). This angle is in radians and is greater than \(\pi\), so it lies in the third quadrant since \(\pi < 7\pi/6 < 3\pi/2\).

Find the reference angle for \(7\pi/6\). The reference angle \(\theta_r\) is the acute angle formed with the x-axis. For angles in the third quadrant, \(\theta_r = \theta - \pi\). So, calculate \(\theta_r = 7\pi/6 - \pi = \pi/6\).

Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). To find \(\csc 7\pi/6\), we need to find \(\sin 7\pi/6\) first.

Determine the sign of \(\sin 7\pi/6\). Since \(7\pi/6\) is in the third quadrant, where sine is negative, \(\sin 7\pi/6 = -\sin \pi/6\).

Use the known exact value \(\sin \pi/6 = \frac{1}{2}\). Therefore, \(\sin 7\pi/6 = -\frac{1}{2}\), and so \(\csc 7\pi/6 = \frac{1}{\sin 7\pi/6} = \frac{1}{-\frac{1}{2}}\).

Risposta video verificata per un problema simile:

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

Durata del video:

Concetti chiave

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

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to an angle between 0 and 90 degrees (or 0 and π/2 radians). Using reference angles allows you to find exact trigonometric values without a calculator.

Video consigliato:

Percorso guidato

5:31

Reference Angles on the Unit Circle

Trigonometric Functions and Their Signs in Different Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. For example, cosecant (csc) is positive in quadrants where sine is positive. Knowing the quadrant of the given angle helps determine the correct sign of the trigonometric value after using the reference angle.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Exact Values of Common Angles

Certain angles like π/6, π/4, and π/3 have well-known exact trigonometric values. For instance, sin(π/6) = 1/2, so csc(π/6) = 2. Recognizing these standard values is essential for finding exact answers without a calculator when working with reference angles.

Video consigliato:

Percorso guidato

3:47

Introduction to Common Polar Equations

Pratica correlata

Domanda del libro di testo

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.

6 3 2 3 6 6 3 2 3 6

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

<IMAGE>

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.

sec 3𝜋/2

450

views

Domanda del libro di testo

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. - 38𝜋/9

668

views

Domanda del libro di testo

Convert 135° to an exact radian measure.

1045