Question

How do you write an equation of the cosine function with amplitude 3 and period $4\pi ?$

Answer 1

Hint: This problem deals with finding the period of the given function. The period of a periodic function is the interval between two matching points on the graph. In other words, it is the distance along the x-axis that the function has to travel before it starts to repeat its pattern. The basic sine and cosine functions have a period of $2\pi $, while tangent has a period of $\pi $.

Complete step-by-step solution:
Given that there is a cosine function with amplitude 3 and with a period of $4\pi $.
We know that the period of cosine trigonometric function is $2\pi $.
The standard form of a function, equation is given by:
$ \Rightarrow y = f(x)$
Here $f(x) = A\cos \left( {Bx - C} \right) + D$
$ \Rightarrow y = A\cos \left( {Bx - C} \right) + D$
Here $A$ is the amplitude of the function.
Here the period, $P$ of the cosine trigonometric standard function is given by:
$ \Rightarrow P = \dfrac{{2\pi }}{B}$
$\therefore B = \dfrac{{2\pi }}{P}$
Here given that the period of the cosine function is $4\pi $, hence substituting it in place of $P$.
$ \Rightarrow B = \dfrac{{2\pi }}{{4\pi }}$
$\therefore B = \dfrac{1}{2}$
Given that the amplitude of the function is 3, hence the value of $A$ is given by:
$\therefore A = 3$
Hence writing the equation, which is given below:
$ \Rightarrow y = 3\cos \left( {\dfrac{1}{2}x - 0} \right) + 0$
As in the given information nothing is mentioned about $C$ and $D$.
$\therefore y = 3\cos \left( {\dfrac{x}{2}} \right)$

The equation of the cosine function with amplitude 3 and period $4\pi $ is $y = 3\cos \left( {\dfrac{x}{2}} \right)$.

Note: Please note that the fundamental period of a function is the period of the function which are of the form, $f\left( {x + k} \right) = f\left( x \right)$ and $f\left( x \right) = f\left( {x + k} \right)$, then $k$ is called the period of the function and the function $f$ is called a periodic function. The period is the length of the smallest interval that contains exactly one copy of the repeating pattern. Any part of the graph that shows this pattern over one period is called a cycle.