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Select the sinusoidal function sine or cosine, enter the values, and Click “Calculate” to find the amplitude and period of a function.

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This calculator determines the amplitude, period, phase shift, and vertical shift for a periodic sinusoidal function, such as sine (f(x)=A⋅sin(Bx−C)+D) and cosine (f(x)=A⋅cos(Bx−C)+D).

Understanding the amplitude and period is important because they help to model a sinusoidal function’s patterns over time.

It is half of the distance between the crest and trough on a sinusoidal wave. For a sine and cosine function, its value is ‘1’ as the centerline is ‘0’ and the range of the function is \(\left(-1, 1\right)\)

The length of one complete cycle of a periodic wave is called its period. For sinusoidal functions (sine & cosine), the fundamental period is \(2\pi\), since both functions repeat their pattern after this period.

- \(sin\left(0\right) = sin\left(2\pi\right) = sin\left(4\pi\right) = …\)
- \(cos\left(0\right) = cos\left(2\pi\right) = cos\left(4\pi\right) = …\)

Phase shift is the horizontal movement of an entire wave function in either the left or right direction. It does not affect the shape, amplitude, and period of a function, but it shifts the whole wave pattern in a particular x-axis direction. You can figure out the horizontal shift (movement) of the wave, which is relative to its standard position for a function with the phase shift calculator.

As the name suggests, vertical shift moves the entire sinusoidal function up or down. It also does not affect the amplitude, period, and overall pattern of the wave.

Methods to determine amplitude and period are as follows:

Either its manual calculation or the use of amplitude and period calculator, these equations can be considered:

\(y = A \sin\left(Bx + C\right) + D\) or \(y = A \cos\left(Bx + C\right) + D\)

3.1.1. Amplitude:

\(\text{Amplitude} = A\)

3.1.2. Period:

\(B = \frac{2\pi}{|B|}\)

3.1.3. Phase Shift:

\(\text{Phase Difference} = -\dfrac{C}{B}\)

3.1.4. Vertical Shift:

\(\text{Vertical Shift} = D\)

This method works if you have the graph and can analyze it to determine values:

3.2.1. Amplitude:

- Identify the mean line and peak of the wave
- Calculate the distance between these two points
- The recorded value will be the amplitude

3.2.2. Period:

- Identify two consecutive peaks of the wave
- Calculate the horizontal distance between these two peaks
- The result will be the period

How to find amplitude and period for the following sinusoidal function:

\(y = 3 \sin\left(5x + 1\right) + 9\)

Solution:

**Step 1: **

Find Amplitude

\(\text{Amplitude} = A\)

\(\text{Amplitude} = 3\)

**Step 2:**** **

Calculate the Period

\(B = \frac{2\pi}{|B|}\)

\(B = \frac{2\pi}{|5|}\)

\(B = \dfrac{6.28}{|5|}\)

\(B = 1.256\)

**Step 3:**** **

Find Phase Shift

\(\text{Phase Shift} = -\dfrac{C}{B}\)

\(\text{Phase Shift} = -\dfrac{1}{1.256}\)

\(\text{Phase Shift} = -0.796\)

**Step 4: **

Find Vertical Shift

\(\text{Vertical Shift} = D\)

\(\text{Vertical Shift} = 9\)

Yes, as it defines the distance between mean and peak points, which is always positive. Our amplitude calculator can determine the amplitude of your sinusoidal function, considering both positive and negative values.

The amplitude of zero does not exist because the zero function represents a flat line that overlaps the equilibrium line. In this condition, the value of B becomes zero, which is not a trigonometric function.

No! tan(x) is not a periodic function due to the following reasons:

- The graph of tan(x) has vertical asymptotes at every odd multiple of π/2 (pi/2, 3π/2, 5π/2, etc.). Sinusoidal functions don't have any holes or gaps in their graphs and their curves are always smooth
- The range of sine and cosine is (-1, 1). But a tangent function has a range in all real numbers

However, tan(x) is a periodic function because it has a period of \(\pi\).

Wikipedia: Amplitude, Peak amplitude & semi-amplitude, Peak-to-peak amplitude, Pulse amplitude, Amplitude normalization

Khan Academy: Midline, amplitude, and period

Lumen Learning: Amplitude and wavelength

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