# Simple harmonic motion proof

1. Dec 18, 2009

### lmedin02

1. The problem statement, all variables and given/known data
Prove that there exists a number A>0 and \phi such that acos(ct)+bsin(ct)=Acos(ct-\phi).

2. Relevant equations
a,b,c are predermined constants where c>0. From this equation I can justify conclusions regarding the amplitude, frequency, and so forth of a simple harmonic ocillator.

3. The attempt at a solution
Obviously if a (or b) is 0, then A is equal b (or a, respectively) and \phi is 0. Thus, I can now assume that a and b are not 0. I try defining two different functions and proving that they are equal for every t using properties of the derivatives.

Last edited: Dec 18, 2009
2. Dec 18, 2009

### rock.freak667

expand out Acos(ct-φ) and then equate coefficients. You should get two equations. Just try to relate them.

Hint: sin2x+cos2x=1

3. Dec 18, 2009

### lmedin02

Got it. I expanded using the trig sums of angles formula for cosine. Thank you.

4. Dec 19, 2009

### Gregg

how do you finish this?

do you get b=-Asin(phi) and a = Acos(phi)

then phi = arctan(b/a)

and A = a/(cos(phi))

do you say that there exists phi = arctan(b/a) > 0 which implies cos(phi) > 0 for 0<phi<pi/4. provided that a > 0 A > 0. etc? I dont see how you 'prove' this.

5. Dec 19, 2009

### rock.freak667

you'd get b=Asinφ and a = Acosφ

consider what a2+b2, gives. Since tanφ=b/a, then φ exists since a,b≠0