Find all solutions in the interval [0, 2π).

2 sin^2 x = sin x

Question

Find all solutions in the interval [0, 2π).

2 sin^2 x = sin x

mathmale · Answer

I'd suggest you re-write the given equation first as:$$2 \sin^2 x = \sin x$$ and next as$$2 \sin^2 x-\sin x=0$$ and lastly as$$2 (\sin x)^2-\sin x=0$$

mathmale · Answer

Try factoring that.  Just factor out sin x.  Can you now find the two solutions (values of sin x)?

Once you have those, solve for the angles x on the interval [0, 2Pi].

anonymous · Answer

Umm I'm going t  otry to factor it out first

anonymous · Answer

sinx (sinx -1) ? I get a little confused wit factoring and sin's

anonymous · Answer

@mathmale

mathmale · Answer

OK.  Let's take a different approach to make this problem look more familiar.

anonymous · Answer

alrighty

mathmale · Answer

We start with $$2 (\sin x)^2-\sin x=0$$and substitute w for sin x:

mathmale · Answer

$$2w^2-w=0$$

anonymous · Answer

2(w)^2 -w=0

mathmale · Answer

We factor that and get what factors?

anonymous · Answer

Since there isn't three "number"s how do I work it out ? Because I know the trick that you find two digits that when added equal the middle number

anonymous · Answer

w(2w-1)?

anonymous · Answer

@mathmale

mathmale · Answer

Yes.  then either w = 0 or (2w-1)=0.
solve the 2nd equation for w, please.

anonymous · Answer

=1/2

anonymous · Answer

2w+1=0 too?

anonymous · Answer

=-1/2

mathmale · Answer

Good.  Now remember that we let w=sin x a few minutes ago.  

If w= 0, that also means that sin x = 0.  At which values of x within the interval [0,2Pi] is sin x = 0?

anonymous · Answer

pi/2 and 3pi/2

mathmale · Answer

Also, 2w-1=0, or w = +1/2.

this means that sin x = 1/2.  There are two angles within the interval [0,2Pi] at which the sine function = 1/2.  What are those 2 angles?

You should end up with 5 solutions (3 of them come from sin x = 0 and 2 from sin x =+1/2).

I need to get off the 'Net now, but if you're not satisfied that you 've solved this problem completely, leave a note for me, and then someone else or I will respond, later.

mathmale · Answer

I think you mean cos x = 0.  the solutions to that are Pi/2 and 3Pi/2.

But we were working with sin x = 0.  What are the solutions to that?   (there are 3 solutions on the interval [0,2Pi].)

anonymous · Answer

I was looking at the completed unit circle  how do you know if it is cos or sin

anonymous · Answer

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Here are the points on a unit circle. If you want to find when sinx=0 then look where y is 0