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Select the sinusoidal function (sine or cosine), enter the required values, and click "Calculate" to find the amplitude and period of the function. The calculator will display the results with steps.
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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 amplitude and period is important because they help model the patterns of a sinusoidal function over time.
Amplitude is half the distance between the crest and trough of a sinusoidal wave. For standard sine and cosine functions, the amplitude is 1 because the centerline is at 0 and the range of the function is (-1, 1).
The period is the length of one complete cycle of a periodic wave. For sine and cosine, the fundamental period is 2π since the functions repeat their pattern after this interval.
Phase shift is the horizontal movement of a wave left or right. It does not affect the shape, amplitude, or period, but shifts the entire wave along the x-axis. You can calculate this using a phase shift calculator.
Vertical shift moves the entire function up or down along the y-axis. Like phase shift, it does not affect amplitude, period, or overall shape.
Use the general form:
y = A sin(Bx + C) + D or y = A cos(Bx + C) + D
Formulas:
If you have a graph, analyze it as follows:
Find the amplitude, period, phase shift, and vertical shift for:
y = 3 sin(5x + 1) + 9
Step 1: Amplitude
Amplitude = A = 3
Step 2: Period
Period = 2π / |B| = 2π / 5 ≈ 1.256
Step 3: Phase Shift
Phase Shift = -C / B = -1 / 5 = -0.2
Step 4: Vertical Shift
Vertical Shift = D = 9
Yes. Amplitude represents a distance, which is always positive. An amplitude calculator can help determine this value for any sinusoidal function.
A zero function has no amplitude because it represents a flat line. The value of B is zero, so the function does not behave as a standard trigonometric function.
No. Unlike sine and cosine, tan(x) has vertical asymptotes at odd multiples of π/2 and its range is all real numbers. However, it is still periodic with a period of π.
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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