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Periodic Functions: A function is said to be periodic functions if it repeats its values at regular intervals of time.

Example - The trigonometric functions (like sine and cosine) are periodic functions, with period 2π.

The Amplitude is the maximum height from the centerline to the peak (or to the trough). Another way to find amplitude is to measure the height from highest to lowest points and divide that by 2.

Amplitude, in physics, can be defined as the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.

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x = A sin (⍵t + φ)

x is the displacement in meter (m).

A is the amplitude in meter (m).

⍵ is the angular frequency in radians per seconds (radians/s).

T is time in second (s).

φ is the phase shift in radians.

Frequency is the number of occurrences of a repeating event per unit of time.

Frequency is also related to the period.

Relation between amplitude and frequency is given by formula.

Frequency = 1/Period

Let’s understand this with a graph.

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In the above diagram sine function repeats 4 times between 0 and 1.

Hence, the frequency is 4, and the period is 1/4.

The Phase Shift is how far the function is shifted horizontally from the usual position.

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The Vertical Shift is how far the function is shifted vertically from the usual position.

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The generalized equation for a sine graph is given by:

y = A sin (B(x + C)) + D

Where

A is amplitude.

Period is 2π/B.

C is phase shift (positive to the left).

D is a vertical shift.

Graph of the above equation is drawn below:

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Note: Here we are using radian, not degree. Full rotation means 2π radian.

The amplitude period phase shift calculator is used for trigonometric functions which helps us in the calculations of vertical shift, amplitude, period, and phase shift of sine and cosine functions with ease. We have to enter the trigonometric equation by selecting the correct sine or the cosine function and click on calculate to get the results.

Time period is defined as the time taken for one complete cycle of vibration to pass a given point. It is denoted by T. Unit of time period is second.

Formula for time period is 2π/⍵ where ⍵ is angular frequency.

1. y = 2 sin (4(x - 0.5)) + 3

Sol: We will compare the given equation with the standard equation then we will write given value.

So amplitude A = 2

Period = 2π/B. Here B value is 4 . So Period = 2π/4 = π/2.

Phase shift = (-0.5) means it will be shifted to the right by 0.5.

Vertical shift = 3 positive value indicates the centre line is y = +3

Grpah is shown below:

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2. Find frequency of the equation y = 3 sin (100 (t + 0.01)) and draw the graph.

Sol: In this amplitude (A) value is 3.

Period = 2π/B here B value is 100. So Period = 2π/100 = 0.02π

Phase shift = 0.01

We know that Frequency = 1/ Period.

So frequency = 1/0.02π = 50/π.

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FAQ (Frequently Asked Questions)

1. Can an Amplitude Value be Negative?

Ans: No, amplitudes are always positive numbers. The amplitude or peak amplitude of a wave or vibration is a measure of deviation from its central value.

2. What Happens to the Frequency of a Wave if its Time Period Increases?

Ans: As we know frequency is the reciprocal of the time period. So, the frequency decreases when the time period increases.

3. What does a Positive Phase Shift Mean?

Ans: The phase shift of a sine curve is how much the curve shifts from zero. If the phase shift is zero, the curve starts at the origin, but it can move left or right depending on the phase shift. A positive phase shift indicates movement of curve to the left.