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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 11
Capitolo 7, Problema 11

Solve each equation for x, where x is restricted to the given interval.
y = 6 cos x/4 , for x in [0, 4π]

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1
Identify the given equation: \(y = 6 \cos \frac{x}{4}\), and the interval for \(x\) is \([0, 4\pi]\).
Since the equation is in terms of \(y\), decide what value of \(y\) you want to solve for. For example, if you want to find \(x\) when \(y\) equals a specific value, set \(6 \cos \frac{x}{4} = y_0\) where \(y_0\) is that value.
Isolate the cosine term by dividing both sides by 6: \(\cos \frac{x}{4} = \frac{y_0}{6}\).
Use the inverse cosine function to solve for \(\frac{x}{4}\): \(\frac{x}{4} = \arccos \left( \frac{y_0}{6} \right)\).
Multiply both sides by 4 to solve for \(x\): \(x = 4 \arccos \left( \frac{y_0}{6} \right)\). Remember to consider all solutions for \(x\) within the interval \([0, 4\pi]\) by using the periodicity and symmetry of the cosine function.

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Solving Trigonometric Equations

Solving trigonometric equations involves finding all values of the variable that satisfy the equation within a specified interval. This often requires isolating the trigonometric function and using inverse functions or known values of sine, cosine, or tangent to determine solutions.
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How to Solve Linear Trigonometric Equations

Properties of the Cosine Function

The cosine function is periodic with a period of 2π, meaning its values repeat every 2π units. Understanding its graph, symmetry, and key values helps in identifying all solutions within a given interval, especially when the argument of cosine is scaled or shifted.
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Graph of Sine and Cosine Function

Interval Restrictions and Domain Considerations

When solving equations on a restricted interval, it is essential to consider only solutions that lie within that domain. This involves adjusting for the function's period and ensuring that all valid solutions for x fall within the specified range, such as [0, 4π] in this problem.
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Finding the Domain of an Equation
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