Question

Solve the following question in detail:
How do you write an equation of the cosine function with amplitude \[2\], period \[pi\], and phase shift \[\dfrac{{pi}}{2}\]?

Answer 1

Hint: Write the given values. Identify the standard equation of cosine form and deduce the values of amplitude, period and phases shift from the standard equation in terms of variables. Then substitute the given values in the variables to find the individual variables comprising the equation. Then substitute these acquired values of the variables in the standard form to get the final equation.

Complete step-by-step solution:
There is a standard form of the cosine function. We have the values given which we substitute them in the cosine function equation to get the equation. Given values,
Amplitude \[ = 2\]
Period \[ = pi\]\[ = \pi \]
Phase shift \[ = \dfrac{{pi}}{2} = \dfrac{\pi }{2}\]
The standard form of the cosine equation is given as follows;
\[y = a\cos \left( {bx + c} \right) + d\]
Now, we have the formulas for amplitude and period which we derive from the above equation. On the basis of the standard equation of the cosine function, we derive;
Amplitude \[ = \left| a \right|\]
Period \[ = \dfrac{{2\pi }}{b}\]
Phase shift \[ = \dfrac{{ - c}}{b}\]
There is also a vertical shift which is given as \[d\]
\[ \Rightarrow \]Vertical shift \[ = d\]
Now, we are given the actual values of the amplitude, period and phase shift. The vertical shift is considered zero.
Amplitude\[ = \left| a \right|\]\[ = 2\]
Vertical shift \[d = 0\]
Period\[ = \dfrac{{2\pi }}{b} = \pi \]
\[ \Rightarrow b = 2\]
Phase shift\[ = \dfrac{{ - c}}{b} = \dfrac{\pi }{2}\]
Substituting the value from the above acquired period, we get;
\[ \Rightarrow \dfrac{{ - c}}{2} = \dfrac{\pi }{2}\]
Cancelling out the common terms, we get;
\[ \Rightarrow c = - \pi \]
Now, we have the values of \[a,b,c\]and \[d\]
We substitute these values in the standard cosine form to get the final value.
\[y = a\cos \left( {bx + c} \right) + d\]
\[a = 2\]
\[b = 2\]
\[c = - \pi \]
\[d = 0\]
Putting in the formula we get
\[ \Rightarrow y = 2\cos \left( {2x - \pi } \right) + 0\]
Rewriting it in the simplified form, we get;
\[ \Rightarrow y = 2\cos \left( {2x - \pi } \right)\]

Therefore, the equation is \[y = 2\cos \left( {2x - \pi } \right)\]

Note: The period of a function is the time period of one which goes from one peak of the curve to another peak of the curve. Amplitude can be defined as the height from the centre of the line to the peak of the curve. The phase shift is defined as how far the function is shifted horizontally from the usual position and the vertical shift is how far the function is shifted vertically from its original position.