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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 6.3.5
Chapter 7, Problem 6.3.5

Solve for exact solutions over the interval [0, 2π).
cos 2x = -1

Verified step by step guidance
1
Recognize that the equation is \( \cos 2x = -1 \), where the argument of the cosine function is \( 2x \). Our goal is to find all \( x \) in the interval \( [0, 2\pi) \) that satisfy this equation.
Recall the general solution for \( \cos \theta = -1 \). Cosine equals \( -1 \) at \( \theta = \pi + 2k\pi \), where \( k \) is any integer.
Set \( 2x = \pi + 2k\pi \) to match the argument of the cosine function in the equation. This gives the equation \( 2x = \pi + 2k\pi \).
Solve for \( x \) by dividing both sides by 2: \( x = \frac{\pi}{2} + k\pi \).
Find all values of \( x \) within the interval \( [0, 2\pi) \) by substituting integer values of \( k \) and checking which \( x \) values fall into 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 states that cos(2x) can be expressed in terms of x as cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity allows us to rewrite or interpret the equation involving cos(2x) in terms of a single angle x, facilitating the solving process.
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Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values within a given interval that satisfy the equation. For cos(2x) = -1, we determine the angles where the cosine function equals -1, then solve for x by considering the periodicity and domain restrictions.
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Interval and Domain Considerations

When solving trigonometric equations over a specific interval, such as [0, 2π), it is essential to find all solutions within that range. Since the equation involves 2x, the interval for 2x becomes [0, 4π), and solutions must be adjusted accordingly to find all valid x values in the original interval.
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Related Practice
Textbook Question

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


cos 2x = 1/2

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

Answer each question.


Suppose solving a trigonometric equation for solutions over the interval [0, 2π) leads to 2x = 2π/3, 2π, 8π/3. What are the corresponding values of x?

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 (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.

6 sin² θ + sin θ = 1