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.