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Trigonometric functions are a crucial part of mathematics with applications ranging from simple periodic phenomena to complex engineering problems.

At the heart of these functions are the sine, cosine, and tangent, which primarily depend on the measure of angles and the unit circle. Each function has a set of properties that define their behavior, such as period, amplitude, and phase shift.

The most basic trigonometric function, the sine function, denoted as \(y = \sin{x}\), plots a smooth, wave-like curve that oscillates above and below the x-axis, showcasing how a particular angle's sine value changes with the angle itself. It is periodic, meaning it repeats its values in regular intervals, and these intervals are called periods. The standard period for a sine function is \(2\pi\), after which the values start repeating.

The amplitude and frequency of trigonometric functions are essential characteristics that determine the shape of their graphs.

Amplitude

The amplitude of a sine function is the height from the center line of the wave to its peak. This corresponds to the 'stretching' or 'compression' of the wave vertically. In a basic sine function, the amplitude is one, but it can be altered by multiplying the function by a coefficient.

Frequency

Frequency, on the other hand, tells us how often the wave repeats itself over a certain distance along the x-axis. It is inversely proportional to the period of the function. In the function \(y = \sin{nx}\), where \(n\) is an integer, the frequency is \(n/2\pi\). Scaling the function horizontally through multiplying the variable by a number changes its frequency, resulting in a graph that cycles more rapidly (for greater values of \(n\)) or more slowly (for smaller values of \(n\)).

When graphing trigonometric equations such as \(y = \sin{x} - \sin{2x}\), understanding individual sine functions makes it easier to plot their combination.

Individual Graphs

Start by graphing each sine function separately. The graph \(y = \sin{x}\) is straightforward as it represents a basic sine wave with an amplitude of 1 and a period of \(2\pi\). For \(y = \sin{2x}\), the wave will have the same amplitude but twice the frequency, so its period will be \(\pi\). This means that it will complete one full cycle in half the time of \(y = \sin{x}\).

Combining Functions

Following the initial plotting, merge the two graphs by subtracting corresponding y-values of \(\sin{2x}\) from \(\sin{x}\) at each x. This operation could introduce new features into the combined graph, including alterations in amplitude and frequency, and the potential emergence of beats or complex wave patterns.