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
$$\arcsin x$$ $$-1\le x \le 1$$ $$-\frac{\pi}{2}\le y \le \frac{\pi}{2}$$
$$\arccos x$$ $$-1\le x \le 1$$ $$0\le y \le \pi$$
$$\arctan x$$ $$x \in \mathbb{R}$$ $$-\frac{\pi}{2}\lt y \lt \frac{\pi}{2}$$