Amplitude of Function with Sine and CosineDate: 01/11/2004 at 20:14:13 From: Ashley Subject: How do you predict the amplitude of a sine + cosine function How do you predict the amplitude of a function involving both sine and cosine? For instance, how is it possible to predict the amplitude of f(x) = a*sin(x) + b*cos(x) without using an automatic grapher? I know how to predict the amplitude of a certain function such as cosine or sine. But how do you do it when a function is a combination of both types of graphs? Date: 01/11/2004 at 20:47:45 From: Doctor Jerry Subject: Re: How do you predict the amplitude of a sine + cosine function Hello Ashley, You transform the function this way: a*sin(x) + b*cos(x) = sqrt(a^2 + b^2)[(a/sqrt(a^2 + b^2))*sin(x) + (b/sqrt(a^2 + b^2))*cos(x)] The sum of the squares of the two coefficients (a/sqrt(a^2 + b^2)) and b/sqrt(a^2 + b^2) is 1. So, there is an angle t for which cos(t) = a/sqrt(a^2 + b^2) and sin(t) = b/sqrt(a^2 + b^2) So, a*sin(x) + b*cos(x) = sqrt(a^2 + b^2)*sin(x + t). So, it's another sine wave, with amplitude sqrt(a^2 + b^2). - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 01/12/2004 at 02:12:21 From: Ashley Subject: Thank you (How do you predict the amplitude of a sine+cosine function) Thanks Doctor Jerry, you're a lifesaver! |
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