# Find all $x$ that satisfy $3 \cos{2x}=-1$ for $0^{\circ} \leq x \leq 180^{\circ}$ [closed]

Question

Find all $$x$$ that satisfy $$3 \cos{2x}=-1$$ for $$0^{\circ} \leq x \leq 180^{\circ}$$

can you help to solve this? tq

• Help us help you by asking a good question. Here is how to ask a good question - math.meta.stackexchange.com/questions/9959/… Oct 10 at 3:22
– Vega
Oct 10 at 3:39

$$2\cos 4x+3 \cos 2x=-1 \space \space \space \space \space \text{using} \space \space \space (\cos 4x=2\cos^2 2x-1)$$ $$4\cos^2 2x-2+3\cos 2x=-1$$ $$4\cos^2 2x+3\cos 2x-1=0$$ $$\cos 2x=\dfrac{-3\pm\sqrt{3^2-4(4)(-1)}}{2\times4}$$ $$\cos 2x=\dfrac{1}{4},-1$$ Consider the graph of $$\cos 2x$$ or solving one-by-one gives, $$\cos 2x=-1\Rightarrow 2x=\pi\rightarrow x=\dfrac{\pi}{2}$$ We won't consider the other solutions as it is not mentioned in the range above. $$\cos 2x=\dfrac{1}{4}$$ $$2\cos^2 x-1=\dfrac{1}{4}$$ $$\cos^2 x=\dfrac{5}{8}\Rightarrow \cos x=\pm \sqrt {\dfrac{5}{8}}$$ In the interval $$[0, \pi]$$ $$\cos x$$ takes all the values between $$[-1,1]$$ and so both values will occur for any $$x$$ in the interval or you can also check by graph.  • please don't answer questions that have no input from the user's side. we want people to learn, hints are better.
– Vega
Oct 10 at 4:31
• Sorry, will remember @Vega Oct 10 at 4:41
• no problem :) , it was just a heads up, same happened to me once.
– Vega
Oct 10 at 4:48
• Thank you @Vega Oct 10 at 4:49
• off topic, are you a JEE aspirant?
– Vega
Oct 10 at 4:50

Hint: Use $$\cos(2\theta) = 2\cos^2\theta-1$$ and form a quadratic in terms of $$\cos(2x)$$. Find the value of $$\cos(2x)$$ and apply the identity again. Finally solve for $$\cos x$$.