Ch. 3 - Trigonometric Identities and Equations

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

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problema 3.RE.50

Capitolo 3, Problema 3.RE.50

In Exercises 50–53, find all solutions of each equation. cos x = ﹣1/2

Guida verificata passo dopo passo

Recognize that the equation is \(\cos x = -\frac{1}{2}\). We need to find all angles \(x\) where the cosine value is \(-\frac{1}{2}\).

Recall the unit circle values where \(\cos x = \pm \frac{1}{2}\). Specifically, \(\cos x = \frac{1}{2}\) at \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\), so for \(\cos x = -\frac{1}{2}\), the solutions will be in the second and third quadrants.

Identify the reference angle \(\theta\) such that \(\cos \theta = \frac{1}{2}\). This reference angle is \(\theta = \frac{\pi}{3}\).

Write the general solutions for \(\cos x = -\frac{1}{2}\) using the reference angle and the fact that cosine is negative in the second and third quadrants: \(x = \pi - \frac{\pi}{3} + 2k\pi\) and \(x = \pi + \frac{\pi}{3} + 2k\pi\), where \(k\) is any integer.

Simplify the expressions to get \(x = \frac{2\pi}{3} + 2k\pi\) and \(x = \frac{4\pi}{3} + 2k\pi\). These represent all solutions to the equation \(\cos x = -\frac{1}{2}\).

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.

Basic Trigonometric Equations

Solving trigonometric equations involves finding all angles x that satisfy the given equation within a specified domain. For example, solving cos x = -1/2 means identifying all angles where the cosine value equals -0.5, considering the periodic nature of cosine.

Video consigliato:

Percorso guidato

4:34

How to Solve Linear Trigonometric Equations

Unit Circle and Reference Angles

The unit circle helps visualize cosine values as the x-coordinate of points on the circle. To solve cos x = -1/2, one finds the reference angle where cosine is 1/2 and then determines the angles in the appropriate quadrants where cosine is negative (second and third quadrants).

Video consigliato:

Percorso guidato

5:31

Reference Angles on the Unit Circle

General Solution for Cosine Equations

Because cosine is periodic with period 2π, the general solutions for cos x = a are x = ±θ + 2kπ, where θ is the reference angle and k is any integer. This accounts for all possible solutions over the real numbers.

Video consigliato:

Percorso guidato

4:49

Inverse Cosine

Pratica correlata

Domanda del libro di testo

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x = √ 3 sin x

658

views

Domanda del libro di testo

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°

729

views

Domanda del libro di testo

In Exercises 43–44, express each product as a sum or difference. sin 7x cos 3x

701

views

Domanda del libro di testo

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos 2x = -1

527

views

Domanda del libro di testo

In Exercises 35–38, find the exact value of the following under the given conditions:

a. sin(α + β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

783

views

Domanda del libro di testo

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

944

views