Sin2x<Sinx so that x is between 0 and 2Pi
Sinx=0 or cosx=1/2
x=0,Pi/3,Pi, 5Pi/3 and 2Pi
So how do I answer this question from here?
Should I just substitute a value between 0 and Pi/3, Pi/3 and Pi, Pi and 5Pi/3....... and see whether sin2x value is smaller than the sinx value?
Is this a good way to do it or can you suggest a more expedient method to answer this Q? My method is rather time consuming I think - wondering if there's a better methd
Those 5 solutions of x which you found would divide the number line into a few sub intervals . Just substitute a value and check whether its validity .
Example , for (0 , pi/3)
just pick a value in between , and substitute into sinx(2cosx-1) and if its < 0 , then its valid .
Now What I was thinking was of the 2 graphs Sin2x and sinx both graphs drawn on the same (set of axis?)......or both drawn together, and where the sin2x dips below the sinx--->Answer to Question. But when we combine the 2 or whatever to F(x)=Sin2x-Sinx, I can't seem to relate the 2. What i mean is I can't figure out how just because when f(x)<0 it means that the Sin2x graph dips below the sinx graph.
Now I'm really confused. Now I'm wondering why at all I took Sin2x<Sinx as
Sin2x-sinx=0 and found the roots of it!
Can you explain to me whats happening here please?
I wasn't really thinking f(x)=Sin2x-Sinx as a discrete function here, for me to obtain the answer through it or something.
I was just thinking about the 2 graphs Sin2x and Sinx and I was thinking that the regions where the Sin2x dipped below Sinx was the answer to this question.
I can't quite connect the consideration of f(x) here and the checking of whether f(x) is positive..............
Shall be grateful if you or someone else can shed some light on that
In the top diagram of the attached graphic, you'll see these two graphs on the same axes - and you can see where the pink (I hope it's pink on your monitor!) graph, dips below the blue graph, . (I've marked the angles in degrees, because it's easier to show their values that way on my spreadsheet graph.)
In the lower half of the graphic, I've plotted a single graph: . You'll see that this goes below the -axis at exactly the same values as before, where the pink graph was below the blue graph.
It couldn't be simpler really: if , then simply because you're taking away the bigger number ( ) from the smaller one ( ), so you get a negative answer.
Don't make problems for yourself that aren't really there!