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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 6.7
Capitolo 7, Problema 6.7

Solve each equation for x, where x is restricted to the given interval.
y = 5 cos x , for x in [0, π]

Guida verificata passo dopo passo
1
Identify the equation given: \(y = 5 \cos x\), and the interval for \(x\) is \([0, \pi]\).
Since \(y\) is expressed in terms of \(\cos x\), isolate \(\cos x\) by dividing both sides by 5: \(\cos x = \frac{y}{5}\).
Determine the possible values of \(y\) such that \(\cos x = \frac{y}{5}\) is valid, remembering that \(\cos x\) ranges between \(-1\) and \(1\).
Use the inverse cosine function to solve for \(x\): \(x = \arccos\left(\frac{y}{5}\right)\), ensuring that the solutions lie within the interval \([0, \pi]\).
Check if there are any additional solutions within the interval \([0, \pi]\) by considering the properties of the cosine function on this interval, and write down all valid \(x\) values.

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Basic Trigonometric Functions

Trigonometric functions like cosine relate an angle to the ratio of sides in a right triangle or points on the unit circle. Understanding the cosine function's behavior, especially its values between 0 and π, is essential for solving equations involving cos x.
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Solving Trigonometric Equations

Solving equations like y = 5 cos x involves isolating the trigonometric function and finding all angle solutions within the specified interval. This requires knowledge of inverse trigonometric functions and how to interpret multiple solutions in a given domain.
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Domain and Interval Restrictions

The interval restriction (x in [0, π]) limits the possible solutions to those angles within this range. Understanding how to apply domain constraints ensures that only valid solutions are considered when solving trigonometric equations.
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Finding the Domain of an Equation
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