Problem 21 - Precalculus with Calculus Previews

Question

In Problems 21 and 22 , use a graphing utility to obtain the graph of the given function \(f\) on the interval \([0,2 \pi] .\) Use \((2),(3),\) and (4) to write \(f\) in the form \(f\) \((x)=A \sin (B x+\phi) .\) Then find approximate solutions of indicated equation in the interval. \(f(x)=3 \cos 2 x+4 \sin 2 x ; f(x)=5\)

Step-by-Step Solution

Answer

The approximate solutions to the equation are found by solving \(x = \frac{\pi}{2} - \phi) / 2 + k\pi\) within the interval \([0, 2\pi]\).

Step 1: Identify Components of f(x)

Given the function \(f(x) = 3 \cos(2x) + 4 \sin(2x)\), identify the components corresponding to \(a \cos(Bx) + b \sin(Bx)\), where \(a = 3\) and \(b = 4\), and \(B = 2\).

Step 2: Determine Amplitude, A

The amplitude \(A\) can be found using the formula \(A = \sqrt{a^2 + b^2}\). Calculate this as \(A = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

Step 3: Calculate Phase Shift, \(\phi\)

To find the phase shift \(\phi\), use \(\tan \phi = \frac{b}{a}\). Therefore, \(\tan \phi = \frac{4}{3}\). Solving this, \(\phi = \arctan\left(\frac{4}{3}\right)\).

Step 4: Write Function in Desired Form

Using the amplitude and phase shift, rewrite \(f(x)\) as \(f(x) = 5 \sin(2x + \phi)\). Here, \(\phi = \arctan\left(\frac{4}{3}\right)\).

Step 5: Set Equation to f(x) = 5

Given the equation \(f(x) = 5\), substitute into the form we found: \(5 \sin(2x + \phi) = 5\). Simplify to obtain \(\sin(2x + \phi) = 1\).

Step 6: Solve for x in the Interval

The equation \(\sin(2x + \phi) = 1\) implies \(2x + \phi = \frac{\pi}{2} + 2k\pi\). Solve for \(x\) to get \(x = \frac{\pi/2 - \phi}{2} + k\pi\). Use \(k = 0,1\) to find solutions in \([0, 2\pi]\).

Step 7: Find Specific Solutions

For \(k = 0\), \(x = \frac{\pi/2 - \arctan(\frac{4}{3})}{2}\). For \(k = 1\), calculate \(x = \frac{\pi/2 - \arctan(\frac{4}{3})}{2} + \pi\). Any solutions outside \([0, 2\pi]\) are excluded.

Key Concepts

Phase Shift

The concept of phase shift in trigonometric functions refers to the horizontal shift of the graph of a sine or cosine function. When rewriting a function in the form \( f(x) = A \sin(Bx + \phi) \), the term \( \phi \) represents the phase shift.
It's the amount by which the graph is shifted horizontally from the original position. To calculate the phase shift, use the formula:

• \( \tan \phi = \frac{b}{a} \)

In our exercise, this phase shift was calculated using the equation \( \tan \phi = \frac{4}{3} \). By solving, we found \( \phi = \arctan(\frac{4}{3}) \).

Understanding phase shift helps in graphing, as it tells you how much the sine or cosine wave moves left or right. This is crucial when matching the graph to its real-world scenario, such as shifting sound waves or light waves.

Amplitude Calculation

Amplitude in trigonometric functions denotes the peak value or the maximum distance from the central axis. It's a crucial part of the function's equation \( f(x) = A \sin(Bx + \phi) \), where \( A \) represents the amplitude.
The formula to calculate the amplitude is as follows:

• \( A = \sqrt{a^2 + b^2} \)

In our scenario, we have \( a = 3 \) and \( b = 4 \), resulting in \( A = \sqrt{3^2 + 4^2} = 5 \).

Amplitude reflects how high and low the function's waves reach. This impacts the vertical stretch and compress of the graph. Knowing the amplitude is vital in various fields, including physics and engineering, where waveforms are analyzed.

Graphing Utilities

Graphing utilities are essential tools in understanding and visualizing trigonometric functions. With a graphing utility, you can input the function and see its graph over a specified interval, like \([0, 2\pi]\) in our exercise.
These tools allow you to:

• Observe the shape and period of the function.
• Identify the amplitude and phase shift visually.
• Test different values easily to see how they affect the graph.

In our example, using a graphing utility, the function \( f(x) = 3 \cos(2x) + 4 \sin(2x) \) can be represented in a different form, \( f(x) = 5 \sin(2x + \phi) \), to compare against its graphical output.

By visually inspecting the graph, you can confirm the calculations of amplitude and phase shift and understand how they interact to influence the curve's appearance. This interaction is fundamental in fields like bioinformatics and telecommunications, where signal processing and function transformations are key.