4cos(2x) = 8sin(x)cos(x) -- Help with identities

[Vol]

Homework Statement

4cos2x = 8sinxcosx
4cos2x - 8sinxcosx = 0
Now I am stuck. I don't know what identities to use. I can see it was set to 0 for a reason. But why?
I know that answer is
4 - 4tan2x = 0

Relevant Equations

cos2x = cos^2(x) - sin^2(x)?

4cos2x = 8sinxcosx
4cos2x - 8sinxcosx = 0
Now I am stuck. I don't know what identities to use. I can see it was set to 0 for a reason. But why?
I know the answer is
4 - 4tan2x = 0 but how?
Thanks.

 

[SammyS]

Homework Statement:: 4cos2x = 8sinxcosx
4cos2x - 8sinxcosx = 0
Now I am stuck. I don't know what identities to use. I can see it was set to 0 for a reason. But why?
I know that answer is
4 - 4tan2x = 0
Relevant Equations:: cos2x = cos^2(x) - sin^2(x)?

4cos2x = 8sinxcosx
4cos2x - 8sinxcosx = 0
Now I am stuck. I don't know what identities to use. I can see it was set to 0 for a reason. But why?
I know the answer is
4 - 4tan2x = 0 but how?
Thanks.

Do you know an identity for $\sin 2x$ ?

Also, the problem statement does not say what you are to do.
Are you to solve for x ?
The expression you give as the answer does not seem like much of an answer at all. It's more like a step one might take in solving for x .

 

[Vol]

sin2x = 2sinxcosx and yes it's for solving for x. By answer I meant the next step.

 

[SammyS]

sin2x = 2sinxcosx and yes it's for solving for x. By answer I meant the next step.

That is the identity I had in mind. It should be very helpful.

 

[Vol]

Oh, ok. So, the answer is tan2x = 1. I couldn't see you have to divide both sides by cos2x to turn it into tan2x. Thanks y'all.

 

[SammyS]

Oh, ok. So, the answer is tan2x = 1. I couldn't see you have to divide both sides by cos2x to turn it into tan2x. Thanks y'all.

Good !

Now, solve for x .

 

[MidgetDwarf]

In general, when trying to solve an algebraic equation. You want to isolate for the solved variable on one side.

so far you have tan2x=1. But you want to solve for x., ie., x=? What can we do on both sides that would result in x=?

 

[kuruman]

You do nothing on both sides. Just think. What angle has a tangent equal to 1? What is the relation of that angle to x?

 

[vela]

Relevant Equations:: cos2x = cos^2(x) - sin^2(x)?

Generally, you'd want to get everything in terms of functions of either $2x$ or $x$. The identity you listed might be useful if you chose the latter to rewrite the lefthand side, but it would turn out to be more complicated. The alternative is to rewrite the righthand side in terms of $2x$. Then as you discovered, the identity @Office_Shredder pointed you to is useful. As you get more practice, you'll develop an intuition for which way to go and it will feel less like guessing.

4cos2x = 8sinxcosx
4cos2x - 8sinxcosx = 0
Now I am stuck. I don't know what identities to use. I can see it was set to 0 for a reason. But why?

To get from the first equation to the second, $8 \sin x \cos x$ was subtracted from both sides, so you end up with 0 on the righthand side.

Personally, I wouldn't have bothered. I'd have divided both sides by 4 to get $\cos 2x = 2 \sin x \cos x$ and proceeded from there.

 

[jack476]

Divide through by four and use the double-angle formula $\sin(2x) = 2\sin(x)\cos(x)$ to turn the equation into $\cos(2x) = \sin(2x)$, then divide through by $\cos(2x)$ and use the identity $\sin(2x)/\cos(2x) = \tan(2x)$ to get $1 =\tan(2x)$; this is the simplified form of the $4-4\tan(2x) =0$ result that you were given. Now check your unit circle and look up which value of $x$ will make $\tan(2x)$ equal to 1, or equivalently, look up the value of $x$ that makes $\cos(2x)$ and $\sin(2x)$ equal to each other.