# Thread: Simple (I think) trigonometric equation

1. ## Simple (I think) trigonometric equation

Well it's been a while since I exercised my trig brain so I'm in need of a bit of help with this one. The problem I'm trying to solve is related to this: Problem 202 - Project Euler and part of it involves solving an equation that might look something like this:

$\sin{(\frac{2\pi}{3}-x)}=1.4 \sin{(x)}$

I actually know what the solution is to this one (by numerical solving) by I don't know how to achieve it algebraically. Any hints anyone could give would be great.

Thanks

2. Originally Posted by spuz
Well it's been a while since I exercised my trig brain so I'm in need of a bit of help with this one. The problem I'm trying to solve is related to this: Problem 202 - Project Euler and part of it involves solving an equation that might look something like this:

$\sin{(\frac{2\pi}{3}-x)}=1.4 \sin{(x)}$

I actually know what the solution is to this one (by numerical solving) by I don't know how to achieve it algebraically. Any hints anyone could give would be great.

Thanks
i don't think pure algebra can get you very far with this. by the addition formula for sine, we have:

$\sin \bigg( \frac {2 \pi}3 - x\bigg) = \frac 75 \sin x$

$\Rightarrow \sin \frac {2 \pi}3 \cos x - \sin x \cos \frac {2 \pi}3 = \frac 75 \sin x$

this simplifies to:

$\frac {\sqrt{3}}2 \cos x - \frac 9{10} \sin x = 0$

and the algebra is not worth it from there. continue numerically

3. Hmm, very interesting. When I typed this into quickmath's equation solver it managed to find an exact solution algebraically though I imagine it must have been through many complicated steps if you say it's not worth the trouble. Oh well, it looks like I may be going about this problem in the wrong way....

4. Continuing from where Jhevon left off, we have $\frac{\sqrt{3}}{2}\cos{x}-\frac{9}{10}\sin{x}=0$
$\frac{\sqrt{3}}{2}\cos{x}=\frac{9}{10}\sin{x}$
$\sin{x}=\frac{5\sqrt{3}}{9}\cos{x}$
$\frac{\sin{x}}{\cos{x}}=\frac{5\sqrt{3}}{9}$
$\tan{x}=\frac{5\sqrt{3}}{9}$
$x=\arctan\frac{5\sqrt{3}}{9}$

I don't see that you can simplify any further than that, except numerically.

--Kevin C.

5. Thanks very much TwistedOne151, that's exactly what I needed. The solution of that equation matches the value of x I found numerically so that's great.