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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.3.29
Capitolo 7, Problema 6.3.29

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
8 sec² x/2 = 4

Guida verificata passo dopo passo
1
Start with the given equation: \(8 \sec^{2} \frac{x}{2} = 4\).
Divide both sides of the equation by 8 to isolate \(\sec^{2} \frac{x}{2}\): \(\sec^{2} \frac{x}{2} = \frac{4}{8} = \frac{1}{2}\).
Recall the identity \(\sec \theta = \frac{1}{\cos \theta}\), so \(\sec^{2} \theta = \frac{1}{\cos^{2} \theta}\). Substitute this into the equation to get \(\frac{1}{\cos^{2} \frac{x}{2}} = \frac{1}{2}\).
Invert both sides to solve for \(\cos^{2} \frac{x}{2}\): \(\cos^{2} \frac{x}{2} = 2\).
Analyze the equation \(\cos^{2} \frac{x}{2} = 2\) and determine if there are any real solutions for \(x\) in the interval \([0, 2\pi)\), considering the range of the cosine function.

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Trigonometric Functions and Their Definitions

Understanding the secant function (sec) is essential; it is the reciprocal of the cosine function, so sec(θ) = 1/cos(θ). This relationship allows converting secant equations into cosine equations, which are often easier to solve.
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Introduction to Trigonometric Functions

Solving Trigonometric Equations

To solve equations like 8 sec²(x/2) = 4, first isolate the trigonometric function, then use algebraic manipulation to find the values of the angle. Recognizing how to handle squared trigonometric functions and applying inverse functions is key.
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How to Solve Linear Trigonometric Equations

Interval and Domain Considerations for Solutions

Solutions must be found within specified intervals, such as [0, 2π) for radians or [0°, 360°) for degrees. Understanding how to find all solutions within these intervals, including using periodicity and symmetry of trig functions, ensures complete and exact answers.
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Finding the Domain of an Equation
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