Hello, Googl!
$\displaystyle \text{Try to write }y \:=\:3\cos x - 4\sin x$
$\displaystyle \text{as: }\;\begin{Bmatrix}y \:=\: R\cos(x-\alpha) \\ \text{and} \\ y \:=\: R\sin(x+\alpha) \end{Bmatrix}$
We have: .$\displaystyle y \;=\;3\cos x - 4\sin x $
Multiply by $\displaystyle \frac{5}{5}\!:$
. . $\displaystyle y \;=\;\frac{5}{5}(3\cos x - 4\sin x) \quad\Rightarrow\quad y \;=\; 5\left(\frac{3}{5}\cos x - \frac{4}{5}\sin x\right)$ .[1]
Let $\displaystyle \,\alpha$ be an acute angle is a 3-4-5 right triangle.
Code:
*
/|
/ |
5 / |4
/ |
/α |
*-----*
3
And we have: .$\displaystyle \begin{Bmatrix}\cos\alpha \:=\:\frac{3}{5} \\ \\[-3mm] \sin\alpha \:=\:\frac{4}{5} \end{Bmatrix}$
Substitute into [1]:
. . $\displaystyle y \;=\;5\left(\cos\alpha \cos x - \sin\alpha\sin x)$
Therefore: .$\displaystyle y \;=\;5\cos(x + \alpha)$
. . where $\displaystyle \alpha \:=\:\sin^{\text{-}1}\left(\frac{4}{5}\right) \:\approx\:53.1^o$