Amplitude of Function with Sine and Cosine

Date: 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!