How do Find Amplitude, Period, and Phase Shift?

You can determine the amplitude, period, and phase shift of trigonometric functions easily way! In this post, you will learn about this topic.

How do Find Amplitude, Period, and Phase Shift?

Related Topics

Step-by-step guide to finding amplitude, period, and phase shift


Amplitude refers to the maximum change of a variable from its mean value. The amplitude formula helps determine the \(sin\) and \(cos\) functions. Amplitude is represented by \(A\).

The sine function (or) the cosine function can be expressed as follows:



  • \(x=\) displacement of wave (meter)
  • \(A =\) amplitude
  • \(ω =\) angular frequency (rad/s)
  • \(t =\) time period
  • \(ϕ =\) phase angle

The amplitude formula is also stated as the average of the \(max\) and \(min\) values of the \(sin\) or \(cos\) function. We always take the absolute value of the amplitude.


Phase shift:

Phase shift is the shift when the graph of the sine and cosine function moves from its usual position to the left or right, or we can say how much the function moves horizontally from its usual position in the phase shift.

The phase shift formula for a sinusoidal curve is shown below where the horizontal and vertical shifts are expressed. It can be positive or negative depending on the direction of change from the origin. The phase shift formula can be expressed as:

also, \(\left(F\left(x\right)=A\:sin\:\left(Bx\:−\:C\right)+D\right)\).


  • \((\frac{C}{B}\)) shows the phase shift.
  • \(A\) is the amplitude.


A function \(y = f(x)\) is called a periodic function if there exists a positive real number \(P\) such that \(f(x + P) = f(x)\), for all \(x\) belongs to the real numbers.

The periods of some important periodic functions are as follows:

  • The period of \(sinx\) and \(cosx\) is \(2π\).
  • The period of \(tanx\) and \(cotx\) is \(π\).
  • The period of \(secx\) and \(cosecx\) is \(2π\).

Amplitude, Period, and Phase Shift – Example 1:

Find amplitude, period, and phase shift. \(3\:sin\left(4\left(x\:−\:0.5\right)\right)+5\)


  • amplitude \(A=3\)
  • period \(\frac{2\pi \:}{B}=\frac{2\pi }{4}=\frac{\pi }{2}\)
  • phase shift \(=−0.5\) (or \(0.5\) to the right)

Exercises for Amplitude, Period, and Phase Shift

Find the amplitude, period, and phase shift of each function.

  1. \(\color{blue}{y=115\:sin\:\left(20\:\theta \right)+3}\)
  2. \(\color{blue}{y=3\:cos\:\left(6x+\pi \right)}\)
  3. \(\color{blue}{y=-4\:sin\:\left(\frac{2}{3}x-\frac{\pi }{3}\right)}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{amplitude =115, period =\frac{\pi}{10}}\)
  2. \(\color{blue}{amplitude =3, period =\frac{\pi}{3}, phase\:shift=-\frac{\pi }{6}}\)
  3. \(\color{blue}{amplitude =4, period =3\pi, phase\:shift=-\frac{\pi }{2}}\)

Related to This Article

What people say about "How do Find Amplitude, Period, and Phase Shift? - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.

Leave a Reply

45% OFF

Limited time only!

Save Over 45%

Take It Now!

SAVE $40

It was $89.99 now it is $49.99

The Ultimate Algebra Bundle: From Pre-Algebra to Algebra II