SOLUTION: Solve {{{2cot^2(x)+3csc(x)=0}}} given that 0 &#8804; x < 2&#960;

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Question 1099479: Solve 2cot%5E2%28x%29%2B3csc%28x%29=0 given that 0 ≤ x < 2π
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
2cot%5E2%28x%29%2B3csc%28x%29%22%22=%22%220

Use the identity 1%2Bcot%5E2%28theta%29%22%22=%22%22csc%5E2%28theta%29
solved for cotangent2: cot%5E2%28theta%29%22%22=%22%22csc%5E2%28theta%29-1  

2%28csc%5E2%28x%29%5E%22%22-1%29%2B3csc%28x%29%22%22=%22%220

2csc%5E2%28x%29-2%2B3csc%28x%29%22%22=%22%220

2csc%5E2%28x%29%2B3csc%28x%29-2%22%22=%22%220

Factor:

%28csc%28x%29%5E%22%22%2B2%29%282csc%28x%29%5E%22%22-1%29%22%22=%22%220

csc%28x%29%2B2=0   2csc%28x%29-1=0 
csc%28x%29=-2   2csc%28x%29-1=0
                  2csc%28x%29+=1
                  csc%28x%29=1%2F2

Use the fact that cosecant is the reciprocal of the sine:

sin%28x%29=-1%2F2   sin%28x%29=2

The sine cannot be greater than 1 so we ignore sin(x)=2

matrix%281%2C5%2Cx%2C%22%22=%22%22%2C7pi%2F6%2C%22%2C%22%2C11pi%2F6%29

Edwin