Problem 94
Question
Find the solutions of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your answers. $$\cos 2 x-\cos 6 x=0$$
Step-by-Step Solution
Verified Answer
The solutions within the given interval [0, 2π) are \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).
1Step 1: Equating Given Equations
Given the equation is \(\cos 2 x-\cos 6 x = 0\), it's corresponding to \(\cos 2 x = \cos 6 x\). This relationship implies that the angles under cosine function are equivalent, or differing by \(2n\pi\), where \(n\) is an integer.
2Step 2: Solve for Variable x
Taking the equivalent cases referred to in step 1, we get: \(2x = 6x\) or \(2x = 6x + 2n\pi\), solving these for \(x\), we get \(x = 0\) or \(x = \frac{n\pi}{2}\) respectively. However, we need to ensure that the solution falls within the given interval [0, 2π). For \(x = n\pi/2\), the possible values within the interval are \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).
3Step 3: Verification with Graphing Utility
Plotting both functions \(y = \cos(2x)\) and \(y = \cos(6x)\) will show intersections of the curves at \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\) on the interval [0, 2π) which confirm our solutions.
Key Concepts
Understanding the Cosine FunctionGraphing Utility VerificationSolving Trigonometric Identities
Understanding the Cosine Function
The cosine function is a fundamental concept in trigonometry, essential for solving various mathematical problems. It represents the ratio of the adjacent side to the hypotenuse of a right-angled triangle. The cosine function, denoted as \( \cos \), is also a periodic function, meaning it repeats its values in regular intervals called periods. For cosine, the period is \(2\pi\).
In the context of the equation \(\cos 2x - \cos 6x = 0\), we see two cosine functions with different arguments. Analyzing these functions requires understanding that they exhibit wave-like behaviors, cresting and troughing symmetrically above and below the horizontal axis.
To graph these functions, each point along the horizontal axis represents an angle measure, with the corresponding vertical value being the cosine of that angle. In solving \(\cos 2x - \cos 6x = 0\), we equate the two functions and seek the angle \(x\) where they have the same value, indicating their wave crests or troughs coincide.
In the context of the equation \(\cos 2x - \cos 6x = 0\), we see two cosine functions with different arguments. Analyzing these functions requires understanding that they exhibit wave-like behaviors, cresting and troughing symmetrically above and below the horizontal axis.
To graph these functions, each point along the horizontal axis represents an angle measure, with the corresponding vertical value being the cosine of that angle. In solving \(\cos 2x - \cos 6x = 0\), we equate the two functions and seek the angle \(x\) where they have the same value, indicating their wave crests or troughs coincide.
Graphing Utility Verification
Verification using a graphing utility is a modern approach to validating solutions obtained in trigonometry. After solving an equation analytically, these utilities can provide a visual confirmation of the results.
For the equation \(\cos 2x - \cos 6x = 0\), plotting \(y = \cos(2x)\) and \(y = \cos(6x)\) on a graphing calculator or software will display two curves. By examining where these curves intersect, we can determine the angles that satisfy both functions simultaneously. The intersection points correspond to the solutions of the original equation within the given interval \( [0, 2\pi)\).
This graphical verification is particularly useful for students because it not only confirms their answers but also enhances their understanding of the behavior of trigonometric functions and their intersections.
For the equation \(\cos 2x - \cos 6x = 0\), plotting \(y = \cos(2x)\) and \(y = \cos(6x)\) on a graphing calculator or software will display two curves. By examining where these curves intersect, we can determine the angles that satisfy both functions simultaneously. The intersection points correspond to the solutions of the original equation within the given interval \( [0, 2\pi)\).
This graphical verification is particularly useful for students because it not only confirms their answers but also enhances their understanding of the behavior of trigonometric functions and their intersections.
Solving Trigonometric Identities
Trigonometric identities are equations that hold true for all values within the domain of the trigonometric functions involved. These identities are tools that can simplify complex trigonometric equations, making them easier to solve.
In the given exercise, the trigonometric equation \(\cos 2x - \cos 6x = 0\) does not immediately appear to be an identity; however, it is an equation that allows for the application of identities to find solutions. By recognizing that cosine is an even function—a function for which \(\cos(-\theta) = \cos(\theta)\)—we can consider angles that differ by multiples of the period \(2\pi\) as potential solutions.
Solving such an equation typically involves isolating the trigonometric terms, using algebraic techniques, and applying known identities to find angles \(x\) that satisfy the equation over the specified interval. This process requires careful consideration of all possible solutions within the domain and is foundational for students' growth in trigonometry.
In the given exercise, the trigonometric equation \(\cos 2x - \cos 6x = 0\) does not immediately appear to be an identity; however, it is an equation that allows for the application of identities to find solutions. By recognizing that cosine is an even function—a function for which \(\cos(-\theta) = \cos(\theta)\)—we can consider angles that differ by multiples of the period \(2\pi\) as potential solutions.
Solving such an equation typically involves isolating the trigonometric terms, using algebraic techniques, and applying known identities to find angles \(x\) that satisfy the equation over the specified interval. This process requires careful consideration of all possible solutions within the domain and is foundational for students' growth in trigonometry.
Other exercises in this chapter
Problem 93
(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi
View solution Problem 93
Verify that for all integers \(n, \sin \left[\frac{(12 n+1) \pi}{6}\right]=\frac{1}{2}.\)
View solution Problem 94
Use the figure, which shows two lines whose equations are \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2}\). Assume that both lines have positive slopes. Deri
View solution Problem 94
Use a graphing utility to solve the equation for \(\theta,\) where \(\mathbf{0} \leq \boldsymbol{\theta}
View solution