Application for Addition Formulae

$$y\ =\ 2\ \sin\ \theta+\cos\theta$$  

What is the modulus (R) of the above expression?

How difficult was it for you to find the modulus in the earlier question?

Which of the following is the correct expression for: $$R\ \cos\ \left(\theta+\alpha\right)$$

After having equate both expressions correctly; it looks something like this. $$2\sin\theta+\cos\theta=\sqrt[]{5}\cos\theta\cos\alpha-\sqrt[]{5}\sin\theta\sin\alpha$$  

How confident are you in comparing the sine and cosine of $$\theta$$  ?

SO what are the correct comparisons for $$\sin\ \theta\ \$$  and $$\cos\theta$$   for the function $$2\sin\theta+\cos\theta=\sqrt[]{5}\cos\theta\cos\alpha-\sqrt[]{5}\ \ \sin\theta\sin\alpha$$  ?

With sine being negative ;  $$\sin\ \alpha=\frac{2}{-\sqrt[]{5}}$$   and cosine being positive;  $$\cos\alpha=\frac{1}{\sqrt[]{5}}$$  , which Quadrant would they  both share ALPHA?