# Thread: find values of p:

1. ## find values of p:

Find the values of p for which the equation $\sin x+p\cos x=2p$ has a solution

2. Hello great_math
Originally Posted by great_math
Find the values of p for which the equation $\sin x+p\cos x=2p$ has a solution
Let $\sin x + p\cos x = r\sin(x+\alpha)=r\sin x\cos\alpha + r\cos x \sin \alpha$

Then $r\cos\alpha = 1$ and $r\sin\alpha = p$

Square and add: $r^2(\cos^2\alpha + \sin^2\alpha) = 1+p^2$

$\Rightarrow r = \sqrt{1+p^2}$

So $\sqrt{1+p^2}\sin(x+\alpha) = 2p$

$\Rightarrow \sin(x+\alpha) = \frac{2p}{\sqrt{1+p^2}}$

And this has solutions provided $-1 \le \frac{2p}{\sqrt{1+p^2}} \le +1$

Can you complete it from here?