How to Calculate the Amplitude of y cos(x) * cos(2x)/2
How to Calculate the Amplitude of y cos(x) * cos(2x)/2
Introduction
Understanding the amplitude of a function is crucial for analyzing its behavior in various mathematical and real-world applications. For the given function y cos(x) * cos(2x)/2, this article presents two effective methods to determine its amplitude.
Method 1: Utilize Trigonometric Identities and Completing the Square
Step 1: Simplify the Function Using Trigonometric Identities
First, we need to simplify the given function using basic trigonometric identities. The relevant identity here is:
[ cos(2x) 2cos^2(x) - 1 ]Using this identity, rewrite cos(2x):
[ y cos(x) * frac{2cos^2(x) - 1}{2} ]Next, simplify the expression:
[ y frac{1}{2} cos(x) * [2cos^2(x) - 1] ] [ y cos^3(x) - frac{1}{2} cos(x) ]Step 2: Complete the Square
Now, we will complete the square for the expression cos^3(x) - 1/2 cos(x).
Let u cos(x). Then, the function becomes:
[ y u^3 - frac{1}{2}u ]To complete the square, observe that:
[ u^3 - frac{1}{2}u left(u - frac{1}{4}right)^3 - frac{1}{64} - frac{1}{2}u ]This can be further simplified to:
[ y left(u - frac{1}{4}right)^3 - frac{1}{64} - frac{1}{2}u ]Substitute u cos(x) back:
[ y left(cos(x) - frac{1}{4}right)^3 - frac{1}{64} - frac{1}{2}cos(x) ]Step 3: Determine the Range to Find the Maximum and Minimum Values
Since cos(x) ranges from -1 to 1, the expression (cos(x) - 1/4)^3 will range from approximately -1/64 to 125/64. Thus, the maximum and minimum values of y can be determined by analyzing these bounds.
The amplitude of the function is the difference between the maximum and minimum values of y.
Method 2: Use of Maxima and Minima through Differentiation
Step 1: Find the Derivative of the Function
The given function is:
[ y frac{1}{2} cos(x) * [2cos^2(x) - 1] ]Let's differentiate it with respect to x:
[ y frac{1}{2} (2cos^2(x) - 1)cos(x) - frac{1}{2}cos^2(x)sin(x) ] [ y' frac{1}{2} left[ 2cos(x) * (-sin(x)) * (2cos^2(x) - 1) - 2cos(x)sin(x) * (2cos^2(x) - 1) right] - frac{1}{2}cos^2(x)cos(x) ]Further simplification gives:
[ y' frac{1}{2} left[ -2cos(x)sin(x)(2cos^2(x) - 1) - 2cos^2(x)sin(x) right] - frac{1}{2}cos^3(x) ]Step 2: Set the Derivative to Zero to Find Critical Points
Set y' 0 and solve for x:
[ -2cos(x)sin(x)(2cos^2(x) - 1) - 2cos^2(x)sin(x) - frac{1}{2}cos^3(x) 0 ]This equation can be solved to find the critical points, which correspond to the maxima and minima.
Step 3: Evaluate the Function at Critical Points and Endpoints
Evaluate the function at the critical points and endpoints x 0, π/2, π, 3π/2, 2π to find the absolute maximum and minimum values of the function. The maximum and minimum values will then give us the amplitude of the function.
Conclusion
Both methods effectively allow us to calculate the amplitude of the function y cos(x) * cos(2x)/2. The first method simplifies the function using trigonometric identities and completing the square, while the second method utilizes the concept of maxima and minima through differentiation.
Understanding these methods is crucial for analyzing trigonometric functions and their properties. Whether you are a student or a professional, mastering these techniques will enhance your problem-solving skills in calculus and beyond.