Expert: Ahmed Salami - 1/11/2009

Question
How do I solve 2 cot^2 2X = 3 csc 2X if 0 </= X < 360 degrees?

Thanks so much for your help!

Answer
Hi Keli,
Right, i'll show you.
Remember, csc^2 y = 1 + cot^2 y
Therefore,
csc^2(2x) = 1 + cot^2(2x)
and
cot^2(2x) = csc^2(2x) - 1
Then, our equation
2cot^2(2x) = 3csc(2x)
becomes
2[csc^2(2x) - 1] = 3csc(2x)
2csc^2(2x) - 2 = 3csc(2x)
2csc^2(2x) - 3csc(2x) - 2 = 0
This is a quadratic equation in csc(2x) which can be solved to give
csc(2x) = 2 or -1/2
resulting in
sin(2x) = 1/2 or -2
but the sine of an angle cant be numerically greater than 1, so we ignore -2
Hence,
sin(2x) = 1/2
2x = arcsin 1/2
  = 30 or 150
x = 15 or 75

Regards