![]() In particular, that gives:Ī * sin(x + 2π) + D = A * sin(x) + D and A * cos(x + 2π) + D = A * cos(x) + D Recall that the sine and cosine functions have periods (no, not that kind of period) equal to 2π, i.e., we have sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x) for any x. Yup, you guessed it: the amplitude of the phase shift equations A * sin(Bx - C) + D and A * cos(Bx - C) + D is simply equal to A. ![]() And indeed, since sin(Bx - C) and cos(Bx - C) are all this time between -1 and 1, the multiplier A changes this range to -1 * A = -A and 1 * A = A. Therefore, the only thing that can affect the amplitude in the phase shift formulas A * sin(Bx - C) + D and A * cos(Bx - C) + D is the non-zero A. That means the centerline falls at D, and the amplitude is still 1 because the values fall as far as 1 away from D. Since the first part gives something between -1 and 1, the whole thing will be between -1 + D and 1 + D (see How to find the vertical shift for comparison). Now let's see what happens if we add D, i.e., if we have sin(Bx - C) + D or cos(Bx - C) + D instead. ![]() In fact, it's because the function f(x) = Bx - C is then a bijection (i.e., a one-to-one correspondence) onto the space of real numbers. What is more, that simple fact doesn't change if we substitute sin(x) or cos(x) for sin(Bx - C) or cos(Bx - C) for a non-zero B and arbitrary C. We know that the sine and cosine functions have values ranging from -1 to 1. First, we show how to find the amplitude. The sections below describe how to calculate each of them based on the notation from the phase shift formula above. In particular, the value is again equal to 0 if we have the two functions unaltered.Īlright, we've learned what the phase shift is, as well as the three accompanying values. In other words, it's the phase shift's twin that concerns the perpendicular direction.
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