What does a sin x + b cos x = R sin(x + α) mean?
Two waves of the same frequency, one a sine and one a cosine, always add to a single sine wave with amplitude R and phase shift α.
// maths › Advanced Applications
Wave Superposition (a sinx + b cosx)
Combine a sin x + b cos x into a single wave R sin(x + α), finding the resultant amplitude and phase shift, in degrees or radians.
a sin x + b cos x = R sin(x + α), R = √(a²+b²), α = arctan(b/a)
Angle unit
Frequently asked questions
How do I find R and α?
R = √(a² + b²) by Pythagoras on the coefficients, and α = arctan(b/a) (using atan2 for the right quadrant).
Can you show a worked example?
3 sin x + 4 cos x: R = √(9+16) = 5 and α = arctan(4/3) ≈ 53.13°, so it equals 5 sin(x + 53.13°). In radians α ≈ 0.9273 rad; the toggle shows it that way.
Why is this useful?
It combines two oscillations into one, which is how alternating currents, sound from two sources, and combined displacements are simplified.
What if a and b are both zero?
Then R = 0 and there is no wave to combine; the calculator flags this.