How this calculation works
Combine a sine term and a cosine term at the same frequency into one sinusoid.
THE FORMULA
a cos x+b sin x=R cos(x−φ), R=hypot(a,b), φ=atan2(b,a)A worked example
3 cos x+4 sin x has amplitude 5 and phase shift about 53.13°.
Good to know
Phase is undefined when both coefficients are zero.
Reference: OpenStax Algebra and Trigonometry (opens in a new tab) ↗
Read the result
Combine a sine term and a cosine term at the same frequency into one sinusoid.
The primary result is labeled “Amplitude”. Keep the input definitions in view when comparing this result with another method.
Try a related calculation
Phase is undefined when both coefficients are zero.
For a related exercise, try the quadratic equation solver.
A few questions, answered
How can I check the default inputs by hand?
3 cos x+4 sin x has amplitude 5 and phase shift about 53.13°.
Which rule does this calculator apply?
a cos x+b sin x=R cos(x−φ), R=hypot(a,b), φ=atan2(b,a)