Trigonometric Equations

**maca404** · January 6th 2013, 06:52 PM

I have been given the problem $3/tanx=2sinx$ I can solve this easily graphically however I am lost algebraically, I think there is squaring and trigonometrical identities involved but this whole thing has really ground me to a halt I have no idea were to start. If someone could give me some pointers that would be great.

**skeeter** · January 6th 2013, 07:35 PM

Originally Posted by maca404

I have been given the problem $3/tanx=2sinx$ I can solve this easily graphically however I am lost algebraically, I think there is squaring and trigonometrical identities involved but this whole thing has really ground me to a halt I have no idea were to start. If someone could give me some pointers that would be great.

$\frac{3\cos{x}}{\sin{x}}=\frac{2\sin^2{x}}{\sin{x} }$ , note $\sin{x} \ne 0$

$3\cos{x}=2(1-\cos^2{x})$

$2\cos^2{x}+3\cos{x}-2=0$

$(2\cos{x}-1)(\cos{x}+2)=0$

Can you finish?

**ibdutt** · January 6th 2013, 07:38 PM

Just do little bit calculations and proceed
3 = 2 sin x tanx = 2 sinx ( sinx / cos x ) = 2 sin^2 x / cosx
So we get 3cosx = 2 sin^2 x
Now convert sin2x in terms of cos^2 x and you have a quadratic in cos x.
I am sure you can now solve it further.

**maca404** · January 6th 2013, 10:11 PM

Ok I see what everyone is getting at now, I will put what I have below but I need to work out the equation from 0 to 2pi. I have the first one worked out not sure how to get the second:

$(2cosx-1)(cosx+2)=0$

$cosx + 2 = 0$

$cosx = -2$

$No Solution$

$2 cosx -1 =0$

$2 cosx = 1$

$cosx = 1/2$

$x = cos^-1(1/2)$

$x = 1.04 rad$

Now I know there is another but I am usure how to find it ?

**grillage** · January 6th 2013, 11:15 PM

If you are working up to $2\pi$ then there will be another angle in the circle that gives $cosx=\frac{1}{2}$ . Your calculator will only give you the one from 0 to $\pi$

**maca404** · January 6th 2013, 11:47 PM

I can see graphically that I can find the next angle by simply doing 2pi - 1.04 , Is this is the corret way to do it ?

**grillage** · January 7th 2013, 12:13 AM

Yes, that sounds good to me.

**skeeter** · January 7th 2013, 02:50 AM

Review your unit circle ...

$\cos{x}=\frac{1}{2}$ at $x=\frac{\pi}{3}$ and $x=\frac{5\pi}{3}$

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