# Setting Up Trigonometric Models

### OVERVIEW

Trigonometric functions are useful for many applications involving periodic behaviour. A few examples include blood pressure, populations that change seasonally, and the phases of the moon. When working with trigonometric models we will use the form below. $y = a \sin(bx -c) + d$

Let's take a look at some of the associated terminology.

For the functions $$y = a \sin(bx -c) + d$$ or $$y = a\cos(bx-c) + d$$,

$\begin{array} aa= \text {the } \textbf{amplitude} \text{(height from midpoint to highest point)} \\ \frac{2\pi}{b} = \text{the } \textbf{period}\\ \frac{b}{2\pi}= \text{the } \textbf{frequency } \text{(number of complete cycles per unit time)} \\ \frac{c}{b} = \text{the } \textbf{phase shift } \text {and} \\ d =\text{the } \textbf{vertical shift} \end{array}$

The following is a graphical representation of the above concepts:

Example: Determine the amplitude and vertical shift of the periodic
function

$$y = 3 \sin(2x-\pi) + 7$$.

Solution:

Amplitude: 3
Vertical Shift: 7

Example: Determine the period of the function $$y = \sin(3x-\pi) + 2$$.

Solution:

Period = $$\frac{2\pi}{b} = \frac{2\pi}{3}$$

Example: How do the frequency and amplitude of the two periodic
functions below compare?

Solution: Function A has a greater amplitude than function B. Function B has a greater frequency than function A.