Question

Find the amplitude, period, phase-shift for $ y = \cos \left( {x + \dfrac{\pi }{2}} \right) $ ?

Answer 1

Hint: According to given in the question we have to determine the amplitude, period and phase-shift for the given trigonometric expression which is $ y = \cos \left( {x + \dfrac{\pi }{2}} \right) $ . So, to determine amplitude, period and phase-shift first of all to determine the amplitude for the given trigonometric expression we have to use the formula to determine the amplitude which is as mentioned below:

Formula used:
 $ \Rightarrow y = A\cos (kx + \psi )$ ------(1)
Where, A is the amplitude, k is the wavenumber $\left(\text{where } k = \dfrac{{2\pi }}{\lambda } \right)$, $ \lambda $ is the wavelength and $ - \dfrac{\psi }{k} $ is the phase-shift.
Obtain the value of the amplitude which can be determined with the help of the formula (A) which is as mentioned in the solution hint.
Now, determine the value of phase-shift which can be determine with the help of the formula (A) as mentioned in the solution hint in which we have to substitute the values in the formula $ - \dfrac{\psi }{k} $ which can help to obtain the required phase-shift.

Complete step by step answer:
First of all to determine the amplitude for the given trigonometric expression we have to use the formula (A) to determine the amplitude which is as mentioned in the solution hint. Hence, on comparing the given trigonometric expression with the formula (A),
 $ \Rightarrow A = 1 $
Where, A is the amplitude.

Now, obtain the value of the amplitude which can be determined with the help of the formula (A) which is as mentioned in the solution hint. Hence, as we can see that these straight off the original equation so values of period can be determined as,
 $
  \Rightarrow \lambda = \dfrac{{2\pi }}{k} \\
  \Rightarrow \lambda = \dfrac{{2\pi }}{1} \\
  \Rightarrow \lambda = 2\pi \\
 $

Now, determine the value of phase-shift which can be determine with the help of the formula (A) as mentioned in the solution hint in which we have to substitute the values in the formula $ - \dfrac{\psi }{k} $ which can help to obtain the required phase-shift. Hence,
 $ \Rightarrow - \dfrac{\psi }{k} = - \dfrac{\pi }{2} $

Hence, with the help of the formula (A) we have determined the amplitude = 1, period $ \lambda = 2 $ , and phase-shift $ - \dfrac{\psi }{k} = - \dfrac{\pi }{2} $ for $ y = \cos \left( {x + \dfrac{\pi }{2}} \right) $ .

Note:
• To obtain the value of period it is necessary that we have to determine the value of k which can be obtain by comparing the expression which is as mentioned in the solution hint and then we have to substitute values in $ \lambda = \dfrac{{2\pi }}{k} $ where, is the period.
• To determine the value of phase-shift it is necessary that we have to determine the value of $ \psi $ and k to substitute these values in the formula $ - \dfrac{\psi }{k} $ to determine phase-shift.