Ch. 6 - Inverse Circular Functions and Trigonometric Equations

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

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 6.3.53

Capitolo 7, Problema 6.3.53

Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate
tan 2x + sec 2x = 3

Guida verificata passo dopo passo

Start with the given equation: \(\tan 2x + \sec 2x = 3\).

Recall the definitions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\). Substitute these into the equation to get \(\frac{\sin 2x}{\cos 2x} + \frac{1}{\cos 2x} = 3\).

Combine the terms over the common denominator \(\cos 2x\): \(\frac{\sin 2x + 1}{\cos 2x} = 3\).

Multiply both sides by \(\cos 2x\) to clear the denominator: \(\sin 2x + 1 = 3 \cos 2x\).

Rewrite the equation as \(\sin 2x - 3 \cos 2x = -1\) and use the identity for a linear combination of sine and cosine: \(a \sin \theta + b \cos \theta = R \sin(\theta + \alpha)\), where \(R = \sqrt{a^2 + b^2}\) and \(\alpha = \arctan(\frac{b}{a})\). Apply this to express the left side as a single sine function and then solve for \(x\) over \([0, 2\pi)\).

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.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For this problem, knowing the relationship between tangent and secant, such as sec²θ = 1 + tan²θ, helps in transforming and simplifying the equation to a solvable form.

Video consigliato:

Percorso guidato

5:32

Fundamental Trigonometric Identities

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the given interval. This often requires algebraic manipulation, use of identities, and considering the periodicity of functions to find all valid solutions between 0 and 2π.

Video consigliato:

Percorso guidato

4:34

How to Solve Linear Trigonometric Equations

Interval and Exact Values

The problem restricts solutions to the interval [0, 2π), meaning all solutions must be found within one full rotation of the unit circle. Solutions should be expressed as exact values (like π/4) or decimal approximations to four decimal places, ensuring clarity and precision in the final answers.

Video consigliato:

Percorso guidato

5:10

Evaluate Composite Functions - Values on Unit Circle

Pratica correlata

Domanda del libro di testo

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

2 cos² x + cos x ― 1 = 0

views

Domanda del libro di testo

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

tan² θ + 4 tan θ + 2 = 0

views

Domanda del libro di testo

cos θ + 1 = 0

views

Domanda del libro di testo

3 csc² x/2 = 2 sec x

views