iouri.gordon 3 years ago Hey guys, I am having troubles solving 1A-6 (b) in Problem Set 1, Single Variable Calculus. Any ideas?

1. dinnertable

Alright so this problem has to do quite a bit with trigonometric identities.To start, write the question as an equation.$sinx - cosx = Asin(x+c)$The first thing that comes to mind is that it's possible to rewrite the right side using the compound angle formula.$sinx - cosx = Asin(x+c)$$sinx - cosx = Asinxcosc + Acosxsinc$From here on it's possible to make a system of equations:$Asinxcosc = sinx$$Acosc = 1$ and$Acosxsinc = -cosx$$Asinc = -1$Now what we're going to do here is make them look like some sort of pythagorean identity so as to simplify them down to one variable.$Acosc = 1$$A^2\cos^2c = 1$and the other equation,$Asinc = -1$$A^2\sin^sc = 1$Now the next part may not be quite obvious, but we can solve for A by adding both of the equations together.$A^2\sin^2c + A^2\cos^2c = 2$$A^2(\sin^2c + \cos^2c) = 2$$A^2(1) = 2$$A=\sqrt{2}$In the first step, we added together both equations which was in equality to 2. Next we just factored out the A squared so that we could use the pythagorean identity to simplify the equation, and solve for A. So now that we have A solved, we need to solve for c. Rewrite the original equation with the A value that we solved.$\sqrt{2}sinxcosc + \sqrt{2}cosxsinc = sinx - cosx$Using the same two system of equations we can now solve for c:$\sqrt{2}sinxcosc = sinx$$\sqrt{2}cosc = 1$$cosc = \frac{ 1 }{ \sqrt{2} }$$c = \frac{ \pi }{ 4 }, \frac{7\pi}{4}$Now for the other equation:$\sqrt{2}cosxsinc = -cosx$$\sqrt{2}sinc = -1$$sinc = \frac{-1}{\sqrt{2}}$$c = \frac{5\pi}{4}, \frac{7\pi}{4}$Because c cannot equal two values, we will use the related angles for both equations. $sinx - cosx = \sqrt{2}sinxcos\frac{7\pi}{4} + \sqrt{2}cosxsin\frac{7\pi}{4}$$sinx - cosx = \sqrt{2}\sin(x + \frac{7\pi}{4})$Although this is right, it is one of the many ways you can express it because sine is periodic. Essentially, this is equivalent to:$sinx - cosx = \sqrt{2}\sin(x-\frac{\pi}{4})$

2. dinnertable

Of course once you understand the main concept of the problem, many shortcuts can be made.

3. dinnertable

Here's a diagram showing the related angles of 7pi/4 and pi/4.

4. anonymous