Re: Trigonometric Equation

My strategy would be to reduce to a single trig function. First, so that . Similarly, . So this equation says

"Clear the fractions" by multiplying through by :

Let y= sin(x) so this becomes

We can write this as

and squaring both sides gives a sixth degree polynomial in y.

Re: Trigonometric Equation

Thanks a lot for your help! It's much appreciated.

I think you may have made two small errors (either that or I misunderstood something)

I think (8y^{2}+6)√(1-y^{2})

should rightly be (8y^{2}-6)√(1-y^{2})

and you also dropped the 2 from 3y(1-2y^{3}) at the end. Unless I made some mistake.

In any case doing things the way you showed me I ended up with

16sin^{6}x-88sin^{4}x+105sin^{2}x-36

which I then factored and got 3 facts, 2 of which were the same, and one of which was impossible (sin(x)=±2 when simplified)

In any case the final answer, if I didn't mess up somewhere along the way, is x=pi/3, 2pi/3, or 4pi/3 in the given interval.

Thanks a lot for your help, I don't know if I ever would have found the answer without you.

Oh and sorry, for some reason I am heaving trouble with latex all of a sudden. I'll look into that later though.

Re: Trigonometric Equation

Could you check that you've copied the equation correctly ?

If that second term were I think it drops out quite nicely.

Re: Trigonometric Equation

I did indeed copy it correctly. The question from a practice exam for my universities entrance exam and I think it not dropping out nicely is expressly their purpose unfortunately.

Re: Trigonometric Equation

Quote:

Originally Posted by

**DCB** Thanks a lot for your help! It's much appreciated.

I think you may have made two small errors (either that or I misunderstood something)

I think (8y^{2}+6)√(1-y^{2})

should rightly be (8y^{2}-6)√(1-y^{2})

and you also dropped the 2 from 3y(1-2y^{3}) at the end. Unless I made some mistake.

In any case doing things the way you showed me I ended up with

16sin^{6}x-88sin^{4}x+105sin^{2}x-36

which I then factored and got 3 facts, 2 of which were the same, and one of which was impossible (sin(x)=±2 when simplified)

In any case the final answer, if I didn't mess up somewhere along the way, is x=pi/3, 2pi/3, or 4pi/3 in the given interval.

Thanks a lot for your help, I don't know if I ever would have found the answer without you.

Oh and sorry, for some reason I am heaving trouble with latex all of a sudden. I'll look into that later though.

Just for information Mathematica is showing an additional solution in this interval at

Re: Trigonometric Equation

Really? Any idea how I could go abut finding that answer? And also unless I am mistaken isn't that the golden ratio?

1 Attachment(s)

Re: Trigonometric Equation

Re: Trigonometric Equation

It certainly is. Nice catch.

Interesting. Ibdutt's cubic is also showing 4 solutions including which turns out to be equal to the golden ratio solution but in a nicer form.