SOLUTION: solve: 3cot^2 3x+1=5csc3x

Question 1167513: solve:
3cot^2 3x+1=5csc3x
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
solve:
3cot^2 3x+1=5csc3x
-------------
Is it cot^2(3x) + 1 ?
or cot^2(3x+1) ?
============================================
3cot^2(3x) + 1 = 5csc(3x)
Multiply by sin^2(3x)
3cos^2(3x) + sin^2(3x) = 5sin(3x)
Sub (1 - sin^2(3x)) for cos^2(3x)
3(1 - sin^2(3x)) + sin^2(3x) = 5sin(3x)
3 - 3sin^2(3x) + sin^2(3x) = 5sin(3x)
2sin^2(3x) + 5sin(3x) - 3 = 0
Sub y for sin(3x)
2y^2 + 5y - 3 = 0
(2y-1)*(y+3) = 0
y = 1/2
sin(3x) = 1/2
3x = pi/6, 11pi/6 + n*2pi -- n = any integer
x = pi/18, 11pi/18 + n*2pi/3
==================================
y = -3
sin(3x) = -3 --- no real solution
x is a complex number, probably not in the scope of the class.

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