Problem 79 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=4 \cos x-\sin (2 x), 0 \leq x \leq 2 \pi$$

Step-by-Step Solution

Answer

Graph the function \(y = 4\cos x - \sin(2x)\) from \(0 \leq x \leq 2\pi\); it combines two trigonometric oscillations.

Step 1: Identify the Given Functions

The problem provides two functions. The first function is \(y_1 = 4\cos x\) and the second function is \(y_2 = -\sin(2x)\). We are asked to add their ordinates, which means adding the values of these functions for each \(x\) on the given interval \(0 \leq x \leq 2\pi\).

Step 2: Express the Sum of Functions

The summed function can be written as \(y = y_1 + y_2 = 4\cos x - \sin(2x)\). We are to graph this function over the interval \(0 \leq x \leq 2\pi\).

Step 3: Use Trigonometric Values

To graph \(y = 4\cos x - \sin(2x)\), evaluate it at key points in the interval \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\) and possibly a few others to understand the behavior, or use graphing software for accuracy.

Step 4: Calculate Example Ordinates

Calculate some example values:- At \(x = 0\), \(y = 4\cdot1 - 0 = 4\).- At \(x = \frac{\pi}{2}\), \(y = 4\cdot0 - 1 = -1\).- At \(x = \pi\), \(y = 4\cdot(-1) - 0 = -4\).- At \(x = \frac{3\pi}{2}\), \(y = 4\cdot0 + 1 = 1\).- At \(x = 2\pi\), \(y = 4\cdot1 - 0 = 4\).

Step 5: Graph the Summed Function

Using calculated points and intermediate values, plot the graph for \(y = 4\cos x - \sin(2x)\). Observe that the graph oscillates and is periodic over the interval \([0, 2\pi]\). The cosine and sine properties make it peak and trough according to their combinations.

Key Concepts

Sum of Functions

When examining trigonometric functions, the Sum of Functions concept means we combine the values of two or more functions to produce a new function. In this particular exercise, we have two original functions:

• The first function is given by \(y_1 = 4\cos x\), which represents a cosine wave scaled by 4.
• The second function is \(y_2 = -\sin(2x)\), a sine wave with a frequency doubled and an amplitude scaled negatively.

To find the sum for \( y = 4\cos x - \sin(2x) \), we add the ordinates (or outputs) of these functions at each given point of \( x \). This results in a singular function that behaves as a composite of the two. The important part of summing functions is to calculate their outputs over the interval, nevertheless keeping track of the individual influences of each function. By graphing, we can visually understand how these functions combine to shape the resulting curve.

Trigonometric Values

Trigonometric values are crucial in graphing trigonometric functions. They serve as references or benchmarks. Some key values that we often evaluate in problems like this include:

• \(x = 0\)
• \(x = \frac{\pi}{2}\)
• \(x = \pi\)
• \(x = \frac{3\pi}{2}\)
• \(x = 2\pi\)

These points are chosen as they represent critical points at which the sine and cosine waves reach their maximum, minimum, or cross the horizontal axis. To better understand the summed function \(y = 4\cos x - \sin(2x)\), we find the function values at these key points:- At \(x = 0\), \(4 \cdot 1 - 0 = 4\)- At \(x = \frac{\pi}{2}\), \(4 \cdot 0 - 1 = -1\)- At \(x = \pi\), \(4 \cdot (-1) - 0 = -4\)- At \(x = \frac{3\pi}{2}\), \(4 \cdot 0 + 1 = 1\)- At \(x = 2\pi\), \(4 \cdot 1 - 0 = 4\)These trigonometric values help outline the behavior of the summed function as it progresses through the interval. The calculated ordinates guide how the graph should oscillate within the defined limits.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals. Trigonometric functions such as sine and cosine are the classic examples of periodic functions.Every cosine and sine function has a specific period at which the entire wave pattern repeats itself. For example:

• The function \(\cos x\) has a period of \(2\pi\).
• The function \(\sin(2x)\) has a reduced period of \(\pi\) due to the frequency doubling.

In our summed function \(y = 4\cos x - \sin(2x)\), the periodic nature plays a critical role in graphing over the interval \(0 \leq x \leq 2\pi\). Within this interval, all waves will repeat at least once, providing a complete view of one cycle of the resulting function. By understanding periodicity, it becomes simpler to predict the overall wave characteristics throughout the cycle, such as where peaks or troughs appear relative to each other. This knowledge helps students to more effectively visualize and sketch periodic combined trigonometric functions.