# What Are Trigonometric Models?

A trigonometric model is used when the data seems to follow a cyclical pattern, repeating over time. Examples of this are the height of the sun in the sky, or changes in temperature over the course of a day. When you want to model your data with trigonometric functions, use one of the formulas below.

Theory

### SineFunction

 $f\left(x\right)=A\mathrm{sin}\left(cx+\varphi \right)+d$

Let $u$ be the $x$-value where the graph rises past the equilibrium for the first time after passing the second ($y-$) axis. This means that $\varphi =-c\cdot u$.

Theory

### CosineFunction

 $f\left(x\right)=A\mathrm{cos}\left(cx+\varphi \right)+d$

Let $u$ be the $x$-value of the first maximum to the right of the second ($y-$) axis. In this case, $\varphi =-c\cdot u$.

These formulas for the rest of the coefficients apply to both the sine and cosine functions:

$\begin{array}{llll}\hfill d& =\frac{{y}_{\mathrm{max}}+{y}_{\mathrm{min}}}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill A& =\frac{{y}_{\mathrm{max}}-{y}_{\mathrm{min}}}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill c& =\frac{2\pi }{P}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$