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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.3.1
Chapter 7, Problem 6.3.1

Solve for exact solutions over the interval [0, 2π).


cos 2x = 1/2

Verified step by step guidance
1
Recognize that the equation is \( \cos 2x = \frac{1}{2} \). Our goal is to find all values of \( x \) in the interval \( [0, 2\pi) \) that satisfy this equation.
Recall that \( \cos \theta = \frac{1}{2} \) at specific standard angles. Specifically, \( \cos \theta = \frac{1}{2} \) when \( \theta = \frac{\pi}{3} \) and \( \theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \).
Set \( 2x = \theta \), so the solutions for \( 2x \) are \( \frac{\pi}{3} + 2k\pi \) and \( \frac{5\pi}{3} + 2k\pi \), where \( k \) is any integer, because cosine is periodic with period \( 2\pi \).
Solve for \( x \) by dividing each solution by 2: \( x = \frac{\pi}{6} + k\pi \) and \( x = \frac{5\pi}{6} + k\pi \).
Find all values of \( x \) within the interval \( [0, 2\pi) \) by substituting integer values of \( k \) (such as 0 and 1) into the expressions and checking which values lie in the interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Double-Angle Identity for Cosine

The double-angle identity expresses cos(2x) in terms of x, commonly as cos(2x) = 2cos²(x) - 1 or cos(2x) = cos²(x) - sin²(x). Recognizing this helps relate the equation cos(2x) = 1/2 to standard cosine values and solve for x.
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Double Angle Identities

Solving Trigonometric Equations

Solving equations like cos(2x) = 1/2 involves finding all angles within the given interval where the cosine value matches 1/2. This requires understanding the unit circle, reference angles, and periodicity of cosine to determine all valid solutions.
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Interval and Domain Considerations

Since the problem restricts solutions to the interval [0, 2π), it is essential to adjust for the double angle by considering the domain of 2x and then mapping solutions back to x within the original interval. This ensures all exact solutions are correctly identified.
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Related Practice
Textbook Question

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.

3 csc x― 2√3 = 0

Textbook Question

Solve each equation for x, where x is restricted to the given interval.

y = 5 cos x , for x in [0, π]

Textbook Question

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.

5 + 5 tan² θ = 6 sec θ

Textbook Question

Solve for exact solutions over the interval [0, 2π).

cos 2x = -1

Textbook Question

Answer each question.


Suppose solving a trigonometric equation for solutions over the interval [0°,360°) leads to 3θ = 180°, 630°, 720°,930°. What are the corresponding values of θ?

Textbook Question

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

sin (x/2) = √2 ― sin (x/2)