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# C3 Question: Cosec(x)=2Cot(x) , 0<x<180 Watch

1. Hi. I'm using two different methods and I get different answers for each method. don't know what the problem is. If we solve it using cosec=1/sin and cot=cos/sin then cos(x)=.5 and x=60 but if we use the following method the answers will be different !

Cosec(x)=2Cot(x) (square each side)
Cosec^2(x)=4Cot^2(x) (Cosec^2=1+cot^2)
1+Cot^2(x)=4Cot^2(x)
Cot^2(x)=1/3
Cot(x)= +1/root3 and -1/root3
Tan(x)= +root3 and -root3
x= +60 and -60
so 60 is one answer which is in first quadrant where tan is positive
since one solution is Tan(x)=-root3, and 0<x<180, so x is in 2nd quadrant where tan is negative. So another solution should be 180-60=120
am I right?
2. You are correct up to x=+60 and x = -60. Check your working from there again. x=120 doesn't work...
3. (Original post by Scott3142)
You are correct up to x=+60 and x = -60. Check your working from there again. x=120 doesn't work...
But tan(-60)=tan(120), so why can't we accept 120? I don't get it
4. I got that there is just one solution which is 60.

Cosec(x) = 2cot(x)

If you multiply through by sin(x)tan(x) you can get to tan(x)=2sin(x)

Then using tan(x)=sin(x)/cos(x) you know that sin(x)/tan(x) = cos(x)

So 2sin(x)/tan(x) = 1

hence 2cos(x) = 1

cos(x) = 0.5

And its in the single positive quadrant of cos and since the range is 0-180 its just 60.

Is that right?

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Updated: December 17, 2010
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