Trig Phase Shift Calculator

Example Data Table

Formula Used

Standard form: y = A f(Bx + C) + D

Phase shift: h = -C / B

Shifted form: y = A f(B(x - H)) + D

Phase shift in shifted form: h = H

Period for sine, cosine, secant, and cosecant: 2π / |B| or 360 / |B|

Period for tangent and cotangent: π / |B| or 180 / |B|

Midline: y = D

Amplitude for sine and cosine: |A|

How To Use This Calculator

Choose the trig function from the list.

Select the equation form that matches your problem.

Enter A, B, C, H, and D values as needed.

Use C for standard form. Use H for shifted form.

Choose degrees or radians before calculating.

Enter an x value when you want one evaluated point.

Press Calculate to view the phase shift and graph data.

Use the CSV or PDF buttons to save results.

Understanding Phase Shift

A trig phase shift shows how far a wave moves left or right. It is not the same as vertical shift. It changes where a cycle starts on the x axis. In real problems, this can describe sound timing, daylight patterns, tides, voltage waves, and repeated motion.

Why The Standard Form Matters

Most classroom equations use y = A f(Bx + C) + D. The calculator also supports y = A f(B(x - H)) + D. These forms look different, but they describe the same horizontal movement. In standard form, the phase shift is -C divided by B. In shifted form, the value H is already the phase shift.

Reading Each Output

Amplitude measures height from the midline for sine and cosine. Tangent and cotangent do not have a true amplitude. Secant and cosecant use a stretch factor instead. Period shows the length of one full cycle. The midline is y = D. A positive phase shift moves the graph right. A negative value moves it left.

Using Degrees Or Radians

Choose degrees when your inputs use 30, 45, 90, or 360. Choose radians when your equation uses pi based values. Keep every coefficient in the same unit system. Mixing units gives a misleading graph and wrong key points.

Advanced Graph Planning

The key point table helps you sketch one cycle. Sine and cosine use quarter period spacing. Tangent uses center and asymptote spacing. Secant and cosecant show where undefined values may appear. The evaluated x value is useful when checking homework or modeling a measurement.

Practical Use

Phase shift is common in engineering, physics, and signal work. It also helps compare two repeating patterns. For example, two waves can have equal period and amplitude, but start at different times. This calculator separates each part, so the equation is easier to understand.

Accuracy Tips

Use exact coefficients when possible. Avoid rounding B too early, because period and phase shift both depend on it. Review the expanded C value when converting from shifted form. That check prevents sign errors. Export your result when you need a saved record for notes, worksheets, or reports. Small changes in C can move the graph a lot when B is large. Always compare the shift with the period too.

FAQs