# How do you write an equation of the sine function with amplitude 5, period 3pi, and phase shift –pi?

Mar 20, 2017

$y = 5 \sin \left(\frac{2}{3} x + \frac{2 \pi}{3}\right)$

#### Explanation:

The standard form of the $\textcolor{b l u e}{\text{sine function}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a \sin \left(b x + c\right) + d} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where amplitude "=|a|, "period } = \frac{2 \pi}{b}$

$\text{phase shift "=-c/b" and vertical shift } = d$

We have to find a, b and c

• a=5to"given"

• (2pi)/b=3pirArrb=(2pi)/(3pi)=2/3

• -c/(2/3)=-pi

$\Rightarrow \frac{3}{2} c = \pi \Rightarrow c = \frac{2 \pi}{3}$

$\Rightarrow y = 5 \sin \left(\frac{2}{3} b + \frac{2 \pi}{3}\right) \leftarrow \text{ is the equation}$