Textbook Question
Solve for exact solutions over the interval [0, 2π).
cos 2x = 1/2
Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
All textbooks
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.7
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.7Chapter 7, Problem 6.7
Solve each equation for x, where x is restricted to the given interval.
y = 5 cos x , for x in [0, π]
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
6:04
Introduction to Trigonometric Functions
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.
Recommended video:
4:34How to Solve Linear Trigonometric Equations
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.
Recommended video:
3:43Finding the Domain of an Equation
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.
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
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)