# Trigonometric Functions

## Transformations

The graph of \(f(x)+ a\) is the graph of \(f(x)\)

translated \(a\) units along the \(y\)-axis .

The graph of \(f(x+ a)\) is the graph of \(f(x)\)

translated \(-a\) units along the \(x\)-axis.

To understand why it is \(-a\) units along the `x`-axis, consider the equality \(\sin 0=0\). The equation
\(\sin (x+a)=0\) has a root when \(x+a=0 \Leftrightarrow x=-a\). The point \((0,0)\) on the graph of \(\sin x\) corresponds
to the point \((-a,0)\) on the graph of \(\sin (x+a)\).

The graph of \(bf(x)\) is the graph of \(f(x)\)

stretched by a factor \(b\) along the \(y\)-axis.

The graph of \(f(bx)\) is the graph of \(f(x)\)

stretched by a factor \(1/b\) along the \(x\)-axis.

To understand why it is a factor \(1/b\) along the `x`-axis, consider the equality \(\sin 2\pi =0\). The equation
\(\sin (bx)=0\) has a root when \(bx=2\pi \Leftrightarrow x=\frac{2\pi}{b}\). The point \((2\pi,0)\) on the graph of
\(\sin x\) corresponds to the point \(( \frac{2\pi}{b} ,0)\) on the graph of \(\sin (bx)\).

If a periodic function \(f(x)\) has the period \(P\),

then the function \(f(bx)\) has the period \(\frac{P}{b}\).

## Combined transformations

The number \(A\) in the applet above is called the **amplitude** of a sine or cosine function. The
amplitude is half the difference between the maximum and the minimum value of a sine or a cosine function.
The amplitude is never negative. Moving the functions along the `y`-axis by changing the slider B, does
not change the amplitude.

When transforming \(\sin (x)\) to \(A\sin (ax+b)+B\), the stretch by the factor \(\frac{1}{a}\) is done before the graph is translated
\(-b\) units along the `x`-axis. The point \((0,0)\) on the graph of \(\sin (x)\), corresponds to the point
\((\frac{-b}{a},B)\) on the graph of \(A\sin (ax+b)+B\).

When transforming \(\sin (x)\) to \(A\sin (a(x+b))+B\), the graph is translated \(-b\) units along the `x`-axis
before it is stretched by the factor \(\frac{1}{a}\). The point \((0,0)\) on the graph of \(\sin (x)\), corresponds to the point
\((-b,B)\) on the graph of \(A\sin (a(x+b))+B\).

## Inverse trigonometric functions

For en explanation of inverse functions, see Functions - Inverse and Composite Functions.

The inverse functions of sine, cosine and tangent are written \(\arcsin(x)\), \(\arccos(x)\)
and \(\arctan(x)\) respectively. In GeoGebra and most other computer programs they are written
`asin(x), acos(x)`

and `atan(x)`

respectively.

You can restrict the domain of a trigonometric function in infinitely many ways to enable a definition of an inverse function. By convention, however, the inverse functions are defined like this:

function | domain | range |
---|---|---|

\(\arcsin x\) | \(-1\le x \le 1\) | \(-\frac{\pi}{2}\le y \le \frac{\pi}{2}\) |

\(\arccos x\) | \(-1\le x \le 1\) | \(0\le y \le \pi\) |

\(\arctan x\) | \(x \in \mathbb{R}\) | \(-\frac{\pi}{2}\lt y \lt \frac{\pi}{2}\) |

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License