Sine, Cosine and Tangent

Sine and Cosine

The sine and cosine are goniometric functions which are periodical. That means that the function repeats itself after a certain period. The sine and cosine function are used in cases where there is something periodical going on such as vibrations and waves.

sin(x)

The function $sin(x)$ is the function used in mathematics to plot the graph of the sine. This function results in the wave as seen in figure 1 for this graph the domain of $[0,2\pi]$ is used. The standard function $sin(x)$ has a period of $2\pi$ an amplitude of 1 and its starting point is in the origin where the graph is going upwards.

cos(x)

The function $cos(x)$ is the function used in mathematics to plot the graph of the cosine. This function results in the wave as seen in figure 2 for this graph the domain of $[0,2\pi]$ is used. The standard function $cos(x)$ has a period of $2\pi$ an amplitude of 1 and the function has a maximum in the point (0,1) which is its starting position.

Generalized form of sin(x) and cos(x)

A more generalized form of the sine and cosine function and the different variables belonging to it are shown below.

Where A is equal to the y-translation of the whole function, B to the amplitude, C to $\frac{2\pi}{period}$ and D to the x-translation of the function.

Since both the sine and the cosine are waves and the only difference in between them is the x-translation from D, it can be noticed that:

Example

In the figure below an example of a sine of the generalized form is shown.

Tangent

The tangent also is a goniometric function with a period, it can however not be graphed as a wave. The particular thing about the tangent is that it has vertical asymptotes.

tan(x)

The function $f(x) = tan(x)$ can be plotted and this results in the graph as seen below with the domain $[-\frac{1}{2}\pi,\frac{1}{2}\pi]$. The period of the tangent is $\pi$ and it has vertical asymptotes at $x=-\frac{1}{2}\pi$ and $x=\frac{1}{2}\pi$

Generalized form of tan(x)

A more generalized form of the tangent function and the different variables belonging to it are shown below.

Where A is equal to the y-translation of the whole function, B to the 'stretch' of the function, C to $\frac{\pi}{period}$ and D to the x-translation of the function.

Example

In the figure below an example of a tangent of the generalized form is shown with a domain of $[-\frac{1}{2}\pi,\frac{1}{2}\pi]$.