Skip to main content
Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 23
Capitolo 3, Problema 23

Find all solutions of each equation. 3 sin θ + 5 = ﹣2 sin θ

Guida verificata passo dopo passo
1
Start by writing down the given equation: \(3 \sin \theta + 5 = -2 \sin \theta\).
To isolate the sine term, add \(2 \sin \theta\) to both sides to combine like terms: \(3 \sin \theta + 2 \sin \theta + 5 = 0\).
Simplify the left side by combining the sine terms: \(5 \sin \theta + 5 = 0\).
Next, subtract 5 from both sides to isolate the sine term: \(5 \sin \theta = -5\).
Divide both sides by 5 to solve for \(\sin \theta\): \(\sin \theta = -1\).

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:
3m

Concetti chiave

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

Solving Trigonometric Equations

This involves isolating the trigonometric function (like sin θ) and finding all angle values that satisfy the equation within a given domain. Solutions often require algebraic manipulation followed by using inverse trigonometric functions.
Video consigliato:
Percorso guidato
4:34
How to Solve Linear Trigonometric Equations

Properties of the Sine Function

The sine function is periodic with period 2π and ranges between -1 and 1. Understanding its behavior helps determine all possible solutions for sin θ, including those in different quadrants where sine has the same value.
Video consigliato:
Percorso guidato
5:53
Graph of Sine and Cosine Function

Using Inverse Sine and General Solutions

After isolating sin θ, the inverse sine function (arcsin) gives a principal solution. To find all solutions, one must use the general solution formulas θ = arcsin(x) + 2nπ and θ = π - arcsin(x) + 2nπ, where n is any integer.
Video consigliato:
Percorso guidato
4:03
Inverse Sine
Pratica correlata
Domanda del libro di testo

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

887
views
Domanda del libro di testo

In Exercises 1–60, verify each identity. cot² t /csc t = csc t - sin t

897
views
Domanda del libro di testo

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sin 2x = √3 / 2

632
views
Domanda del libro di testo

Find all solutions of each equation. 4 sin θ﹣1 = 2 sin θ

470
views
Domanda del libro di testo

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 𝝅/3 + 𝝅/4 )

908
views
Domanda del libro di testo

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 5𝝅/3 ﹣ 𝝅/4)

797
views