Hello, wiseguy!

I believe this is what they expected for the first one . . .

On your paper, draw an appropriate sketch ? and use it to write

. . $\displaystyle y\:=\:4\cos x+3\sin x$ .as a cosine with a phase displacement.

Divide by 5: .$\displaystyle \dfrac{y}{5} \;=\;\dfrac{4}{5}\cos x + \dfrac{3}{5}\sin x$ .[1]

Let $\displaystyle \theta$ be in this right triangle:

Code:

*
* *
5 * *
* * 3
* *
* θ *
* - - - - - - - - *
4

Then: .$\displaystyle \cos\theta = \dfrac{4}{5},\;\sin\theta = \dfrac{3}{5}$

Substitute into [1]: .$\displaystyle \dfrac{y}{5} \;=\;\cos\theta\cos x + \sin\theta\sin x $

. . . . . . . . . . . . . . .$\displaystyle \dfrac{y}{5} \;=\;\cos(x - \theta)$

. . . . . . . . . . . . . . .$\displaystyle y \;=\;5\cos\left(x - \arcsin\frac{3}{5}\right)$