Find $4\cos\theta-3\sin\theta$, given that $4\sin \theta +3\cos \theta = 5$ Another problem that I already wasted hours on.
Given
$$4\sinθ +3\cosθ  = 5$$
Find 
$$4\cosθ -3\sinθ$$
Help me guys (PS:I'm not that good in maths)

Very important fact of physics and harmonics and digital signal processing:

The sum of two sinusoids of equal frequency is another sinusoid of that same frequency, regardless of amplitude or phase.

Or stated mathematically:

If 
  $$f(x) = A_1\cos\left(\omega\,x + p_1\right) + A_2\cos\left(\omega\,x + p_2\right)$$
  then
  $$f(x) = A_3\cos\left(\omega\,x + p_3\right)$$
  and
  $$A_1 \angle p_1 + A_2 \angle p_2 = A_3 \angle p_3$$
  where $M \angle \theta$ is polar notation.  The same holds is $\cos$ is replaced with $\sin$.

So your problem is:
$$4\sin(\theta) + 3\cos(\theta) = 5$$
$$4\sin(\theta) + 3\sin(\theta + \pi/2) = 5$$
$$M\sin(\theta + p) = 5$$
$$\theta = \arcsin(5/M) - p$$
And you find $M$ and $p$ by adding:
$$M\angle p = 4 \angle 0 + 3 \angle \pi/2$$
$$\begin{cases} M = 5 \\ p = \arctan(3/4)\end{cases}$$
So
$$\theta = \arcsin(1) - \arctan(3/4)$$
$$\theta = \pi/2 + Z\cdot 2\pi - \arctan(3/4)$$
So now
$$y = 4\cos(\theta) - 3\sin(\theta)$$
$$y = 4\cos(\theta) + 3\cos(\theta + \pi/2)$$
$$y = 5\cos(\theta + \arctan(3/4))$$
So
$$y = 5\cos(\pi/2 + Z\cdot 2\pi - \arctan(3/4) + \arctan(3/4))$$
$$y = 5\cos(\pi/2 + Z\cdot 2\pi)$$
$$y = 0$$
