Confused about how to solve this problem combining sinusoids

[guns4monkeys]

Homework Statement

The textbook problems states: y=cos3x +sin3x and you are to combine it to a single cos function and tell what effect the 3 has. The answer in the back of the book is given as sqrt2*cos(3x-pi/4) and that the 3 means the wave has a horizontal dilation of 1/3.

Homework Equations

general solution for a sinusoid y = C+A*cosB(X-D) where C is the axis, A is the amplitude, B is the reciprocal of the period, X is the angle in radians and D is the phase displacement.

The Attempt at a Solution

1. find A using the pythagorean theorem: so from the original equation y=cos3x+sin3x, we get A=sqrt of 1^2+1^2 or the sqrt of 2 (because the coefficients in the above equation are both 1)

2. Find D by finding the inverse tan of 1/1, because the two coefficients in this prob. are both 1. So the inverse tan of 1 is pi/4.

3. Then I substitued pi/4 for D in the general sinusoid equation sqrt of 2 for A, giving me:
y=sqrt2*cosB(x-pi/4)

4. I used the 3's from the original problem as B in my general equation. so now I have:
y=sqrt2*cos3(x-pi/4)
This is what I thought the answer should be...why does the correct answer have the 3 inside the parenthesis like this: y=sqrt2*cos(3x-pi/4)? It seems to me that the 3 should not only be distributed to the X, but should be outside the parenthesis so that it gets distributed over both X and D.

Please help me understand where I'm going wrong.

Thanks so much.

 

[Avodyne]

If you started with y=cos(x)+sin(x), you would get y=sqrt2*cos(x-pi/4), right? Now, to turn y=cos(x)+sin(x) into y=cos(3x)+sin(3x), you replace every x by 3x. You don't replace every x-pi/4 by 3x-3pi/4; that would be equivalent to replacing every x by 3x-pi/2.

 

[guns4monkeys]

Now, to turn y=cos(x)+sin(x) into y=cos(3x)+sin(3x), you replace every x by 3x.

but isn't 3 the reciprocal of the horizontal dilation? ie B in the general sinusoidal equation: y = C+A*cosB(X-D)

If 3 is the reciprocal of the horizontal dilation, how can I simply place it into the (x-D) portion of the equation?

Thanks
guns4monkeys

 

[Avodyne]

I just explained the correct answer using a very general math principle. You've apparently been taught some set of rules for these kinds of combinations. These rules must agree with general math principles, or they're wrong. In general, you're much better off trying to understand the general principle, rather than learning how to apply a set of rules.

 

[guns4monkeys]

I just explained the correct answer using a very general math principle. You've apparently been taught some set of rules for these kinds of combinations. These rules must agree with general math principles, or they're wrong. In general, you're much better off trying to understand the general principle, rather than learning how to apply a set of rules.

I do understand what you said about the substitution of 3x for x in your explanation. Where I am getting confused is the relationship between these two forms of the equation; the general equation form C + A cos B (X-D) where B is the reciprocal of the period and the solution with the 3X in the parenthesis and nothing in the position of B. I really am trying to understand the concepts here. Please be patient with me, I'm not trying to be obnoxious, just trying to really get it.

 

[Avodyne]

I'm sorry, but I don't know what's confusing you; I don't know what you mean by "the solution with the 3X in the parenthesis and nothing in the position of B".

 

[guns4monkeys]

Thanks for trying to help me.
Guns4monkeys