So I have two angle problems...
sin2x = 2cosxcos2x ; which becomes
0 = 2cosxcos2x - sin2x
0 = 2cosxcos2x - 2sinxcosx
0 = 2cosx(cos2x - sin x)
so either
2cosx = 0 OR cos2x - sinx = 0
Thank you.
Originally Posted by CrimesofParis
Okay you did everything correctly up till now. Now you are required to solved for $\displaystyle x$.
The first one $\displaystyle 2\cos x=0$. Divide by two,
$\displaystyle \cos x=0$, now you need to know when $\displaystyle \cos x$ is zero. That is when $\displaystyle x=90,270$ as two non-coterminal angels. To find all solutions it is the same as $\displaystyle x=90+360k$ and $\displaystyle x=270+360k=-90+360+360k=-90+360(1+k)$ where $\displaystyle k$ is any integer. Because as you know that addition and subtraction of 360 keeps the trigonometric function of that angel the same. But $\displaystyle k+1$ is also an integer. Thus, all solutions for $\displaystyle x$ have form. $\displaystyle x=(-1)^{k+1}90+360k, k \epsilon Z $ or in radian form $\displaystyle x=(-1)^{k+1}\frac{\pi}{2}+2\pi k,k \epsilon Z$
For your second question:
$\displaystyle \cos 2x-\sin x=0$
Use the identity
$\displaystyle 1-2\sin^2x=\cos 2x$
Thus,
$\displaystyle 1-2\sin^2x-\sin x=0$
Change signs,
$\displaystyle 2\sin^2x+\sin x-1=0$
Factor,
$\displaystyle (2\sin x-1)(\sin x +1)=0$
Now each one is set to zero,
$\displaystyle 2\sin x-1=0$
Thus,
$\displaystyle \sin x=\frac{1}{2}$
That is when $\displaystyle x=30,150$ as the only two non-coterminal angels. Thus all solution have form $\displaystyle x=30+360k, 150+360k$ But,
$\displaystyle x=30+360k=30+180(2k)$
$\displaystyle x=150+360k=-30+180(2k+1)$
Thus, all solutions have form $\displaystyle x=(-1)^k30+180k$ or in radian form $\displaystyle x=(-1)^k\frac{\pi}{6}+\pi k,k \epsilon Z$
Finally the last factor equal to zero, (of the quadratic)
$\displaystyle \sin x+1=0$
Thus,
$\displaystyle \sin x=-1$ That happens when $\displaystyle x=270$ and that is the only non-coterminal angle. Thus all solutions have form $\displaystyle x=270+360k$. But $\displaystyle 270+360k=-90+360+360k=-90+360(k+1)$.
Thus, all solutions are $\displaystyle x=-90+360k$. Or in radian form $\displaystyle x=-\frac{\pi}{2}+2\pi k,k\epsilon Z$.
If you have difficulty understanding all that manipulating after the solutions just understand that by adding 360 or subtracting 360 you change nothing. Thus, those are all the solutions. All I did was manipulated into a more elegant form.