A sine graph shows y = sin x, repeating along the x-axis with a period of 1.0π. The sine, cosine, and tangent represent ratios of a right triangle, with the sine having a reciprocal called the cosecant. A unit circle can help visualize sine values at different angles. Sine waves are found in math, physics, music, and engineering, and maintain their waveform when combined with another sine wave of the same frequency and phase. In sound, a pure note is represented by a sine wave.
A sine graph is a graph showing the function of y = sin x. A sine wave graph has a function that can also be described as a sine wave. It repeats as it moves along the x-axis, and the cycle it takes for one repetition is known as the period of the sine wave graph. Different analyzes can be made on the period and amplitude of a sine graph, and there are many interesting results that can be drawn from this crucial function.
The sine itself is a measurement given to an angle, representing the ratio of the opposite of the far side to the length of the hypotenuse. It can be contrasted with the cosine, which represents the ratio of the adjacent leg to the hypotenuse, and the tangent, which represents the ratio of the opposite leg to the adjacent leg. Each function also has a reciprocal, for example the cosecant, the reciprocal of the sine, which represents the relationship between the hypotenuse and the opposite leg.
The best way to understand a sine wave graph is to look at a visual representation of a unit circle, which demonstrates where various important sine values fall at different angles radiating from a single circle. It makes it very apparent when sine has the value 0, appearing on the four dots radiating from a cross in the center of the circle, equal to 0.1 or 1.0 or 0 or -1 . This allows us to see that the period of a sine graph equals 1.0π, with each additional period being just one more cycle around the circle.
On a sine graph, this can be seen as a sine wave curving up to a value of 1, then back below the 0 sign to -1, then curving up again to repeat the process . It makes a trough-to-peak traverse every iteration of , and returns to its previous position after 2π. The depression on the Cartesian plane appears, for example, in -π/2 and 3π/2, while the peak appears in -3π/2, π/2. A cosine graph looks very similar to a sine graph, but its peak will appear at, for example, -2π, 0, and 2π.
You can see examples of a sine wave almost everywhere, from pure mathematics to physics to music to electrical engineering. The sine wave is unique in that it maintains the same waveform when another sine wave is added, as long as the second wave has the same frequency and phase. Many basic thought experiments in physics can be demonstrated with a pure sine wave, from a simple pure tone to how a spring oscillates if it’s completely undamped by things like friction.
In sound, a pitch that would appear as a sine wave graph is heard by humans as a pure note. For example, a constant whistling sound usually produces a sine wave when observed in sound recording software. The sound produced by a tuning fork is another good example of a relatively pure sine wave.
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