Problem 73 - Precalculus Essentials

Question

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=\sin x, g(x)=\cos 2 x, h(x)=(f-g)(x)$$

Step-by-Step Solution

Answer

The graph of \(h(x)\) will display the differences between the \(f\) and \(g\) functions according to their \(y\)-coordinates. The calculations necessary to create the \(h\) function's graph involve subtraction of the \(y\)-coordinates of the original functions at each x value in the specified interval.

Step 1: Graph of f(x) =

Firstly, graph \(f(x) = \sin x\). This is a standard sine wave that starts from 0 at \(x = 0\) and has full cycles from 0 to \(2\pi\). This wave rises to 1 at \(\pi/2\) and then falls to -1 at \(\pi\) and again back to 0 at \(2\pi\).

Step 2: Graph of g(x) =

Next, plot \(g(x) = \cos 2x\). This is a cosine wave that starts from 1 at \(x = 0\). However, because of the factor of 2 in the argument, it completes two full cycles from 0 to \(2\pi\).

Step 3: Graph of h(x) =

Finally, plot \(h(x)\) by subtracting \(g(x)\) from \(f(x)\) at each value of x on the interval [0, \(2\pi\)]. This can be done by subtracting the \(y\)-coordinates of \(f(x)\) and \(g(x)\). The points of \(h(x)\) will be the difference in heights of the corresponding points on the \(f(x)\) and \(g(x)\) graphs.

Key Concepts

Sine Wave

The sine wave is a fundamental shape commonly found in mathematics, particularly within the study of trigonometry. It reflects the periodic oscillations of the sine function, \(f(x) = \sin x\), which describes a smooth repetitive wave over a continuous range of angles or times. When graphed in a rectangular coordinate system, the sine wave starts at 0 when \(x=0\), reaches its maximum at \(\frac{\pi}{2}\) with a value of 1, descends to 0 at \(\pi\), reaches its minimum at \(\frac{3\pi}{2}\) with a value of -1, and completes one full cycle at \(2\pi\), returning to 0.

To visualize this, imagine the unit circle where the angle \(x\) is measured from the positive x-axis, the sine of the angle corresponds to the y-coordinate of the point on the circle. As the angle increases, the point moves around the circle, and the y-coordinate goes up and down, creating the wave pattern. This simple yet elegant behavior makes the sine wave incredibly important in modeling naturally occurring phenomena, such as sound and light waves.

Cosine Wave

The cosine wave, described by the function \(g(x) = \cos x\), is another core trigonometric function. However, in the provided exercise, you are working with \(g(x) = \cos 2x\), indicating that the wave's frequency is doubled. This results in the wave completing two full cycles within the same range of \(0\) to \(2\pi\) that a standard cosine wave would complete only one.

A cosine wave starts at its maximum value of 1 when \(x = 0\) and follows a similar pattern as the sine wave but with a phase shift of \(\frac{\pi}{2}\) to the left. In practice, the cosine wave represents the same unit circle motion as the sine wave but corresponds to the x-coordinate of the point on the unit circle as the angle \(x\) increases. This relationship between sine and cosine functions is an example of the phase shift property used in modeling alternating currents and other phenomena where timing or starting position shifts the behavior of the wave.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their respective domains. These identities are used to simplify expressions, solve equations, and convert between different trigonometric functions. Some fundamental identities include the Pythagorean identities, the angle sum and difference identities, and the double-angle identities.

For instance, in the context of the provided exercise, understanding how the trigonometric identities work could help in manipulating the expressions for \(f(x) = \sin x\) and \(g(x) = \cos 2x\), especially when the exercise requires you to handle \(h(x) = (f - g)(x)\). Knowing how sine and cosine relate to each other can simplify the graphing process and elucidate the resulting wave interactions. This demonstrates why familiarity with these identities is not just academic but can also be very practical.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane defined by two perpendicular axes: horizontally we have the x-axis, and vertically the y-axis. Every point in the plane can be represented by a unique pair of coordinates \( (x, y) \) corresponding to its horizontal and vertical positions. This system is a foundational tool in graphing equations and functions, including trigonometric functions like sine and cosine waves.

The exercises you encounter will often require graphing within this system, since it allows for precise representation of the relationship between two variables. The system's grid layout enables easy visualization of patterns and behaviors of functions, such as the repeating cycles of trigonometric waves. When graphing \(f(x) = \sin x\), \(g(x) = \cos 2x\), and \(h(x) = (f - g)(x)\) within this coordinate system, observing how the two original waves combine to form the resulting graph \(h(x)\) becomes an insightful exercise in understanding wave interactions.