Reduce the equation to a single cosine function with positive amptitude

Question

3cos(8pi*x)-8sin(8pi*x)

-3cos(8pi*x)-8sin(8pi*x)

8sin(8pi*x)-3cos(8pi*x)

3cos(8pi*x)+8sin(8pi*x)

Answer 1

Given:

3 cos( 8πx ) - 8 sin( 8πx )

-3 cos( 8πx ) - 8 sin( 8πx )

8 sin( 8πx ) - 3 cos( 8πx )

3 cos( 8πx ) + 8 sin( 8πx )

Find:

A cos( 8πx + φ )

A = ?

φ = ?

For all 4 cases

Solution:

Since I don't have the time to do every solution, I'll give you the framework and the first solution and the rest should follow

The goal is to set the equation I've displayed in the Find section to be equivalent to each sum or difference of sine and cosine like so:

A cos( 8πx + φ ) = 3 cos( 8πx ) - 8 sin( 8πx )

What this allows us to do is that we can expand the left hand side of the equation using the Cosine Angle Sum formula:

cos( F + G ) = cos F * cos G - sin F * sin G

So let's do it:

cos( 8πx ) * A cos( φ ) - sin( 8πx ) * A sin( φ ) = 3 cos( 8πx ) - 8 sin( 8πx )

Thus: A cos( φ ) = 3 A sin( φ ) = 8

Note: KEEP WATCH OF SIGNS ON THE TRIG TERMS, if the right hand side was instead "+ 8 sin( 8πx )", then A sin( φ ) would equal -8 instead of +8

[ A cos( φ ) ]² + [ A sin( φ ) ]² = 3² + 8² = A² = 73 ∴ A = √( 73 ) ≈ 8.544

A sin( φ ) / A cos( φ ) = 8 / 3 = tan φ ∴ φ = tan^-1( 8/3 ) ≈ 1.212 rad

So, 3 cos( 8πx ) - 8 sin( 8πx ) = √( 73 ) cos( 8πx + tan^-1( 8/3 ) ) ≈ 8.544 cos( 8πx + 1.212 )

I hope this helps! Sorry that I couldn't post all the problems solutions, I would have eventually hit the character limit on Wyzant anyway! Message me in the comments if you have any questions, comments, or concerns on what and how I did above!