Question

How do you write the equation of a cosine function: Amplitude$= \dfrac{2}{3}$, Period $= \left( {\dfrac{\pi }{3}} \right)$ , Phase shift $= \left( { - \dfrac{\pi }{3}} \right)$ and Vertical shift $= 5$ ?

Answer 1

Hint : In the given question, we are required to find the equation of a cosine function whose amplitude is $\left( {\dfrac{2}{3}} \right)$ , period is $\left( {\dfrac{\pi }{3}} \right)$ , phase shift is $\left( { - \dfrac{\pi }{3}} \right)$ and vertical shift is $5$ . One should know the meaning of these parameters and terms given to us in the question in order to solve such types of problems.

Complete step-by-step answer :
So, we have to find the equation of the cosine function.
We know that a cosine function is basically of the form $A\cos \left( {kx + \phi } \right) + c$ where all the parameters have their own meaning and significance.
In the equation of the cosine function $A\cos \left( {kx + \phi } \right) + c$ , we have amplitude as A, vertical shift as c, phase shift as $\phi$ and period of the cosine function is calculated as $\left( {\dfrac{{2\pi }}{k}} \right)$ .
Now, we are given that the amplitude of the cosine function is $\left( {\dfrac{2}{3}} \right)$ . This means that the value of A in the required cosine equation $A\cos \left( {kx + \phi } \right) + c$ is $\left( {\dfrac{2}{3}} \right)$ .
The period of the cosine function is $\left( {\dfrac{\pi }{3}} \right)$ . This means that the value of $\left( {\dfrac{{2\pi }}{k}} \right)$ is equal to $\left( {\dfrac{\pi }{3}} \right)$ . So, the value of k is $6$ in the required cosine equation $A\cos \left( {kx + \phi } \right) + c$ .
The phase shift of the cosine function is $\left( { - \dfrac{\pi }{3}} \right)$ . So, the value of $\phi$ is $\left( { - \dfrac{\pi }{3}} \right)$ in the required cosine equation $A\cos \left( {kx + \phi } \right) + c$ .
Also, the vertical shift of the graph of cosine function is $5$ . So, the value of c is $5$ .
Now, we put the values of all the parameters into the equation of the cosine function so as to get the required equation.
So, we have, $A\cos \left( {kx + \phi } \right) + c$
$\Rightarrow \left( {\dfrac{2}{3}} \right)\cos \left( {6x - \dfrac{\pi }{3}} \right) + 5$
Opening up the bracket and simplifying the expression further, we get,
$\Rightarrow \dfrac{2}{3}\cos \left( {6x - \dfrac{\pi }{3}} \right) + 5$
So, the required equation of a cosine function: Amplitude$= \dfrac{2}{3}$, Period $= \left( {\dfrac{\pi }{3}} \right)$ , Phase shift $= \left( { - \dfrac{\pi }{3}} \right)$ and Vertical shift $= 5$ is $\dfrac{2}{3}\cos \left( {6x - \dfrac{\pi }{3}} \right) + 5$ .

Note : Cosine is one of the six basic trigonometric functions. Cosine is the ratio of the base to the hypotenuse of a right angled triangle. All these parameters given to us in the question can also be used to sketch a graph of the cosine function as these factors also serve as the graphical transformations.