Solve 2cot(?^(2))x = 5 - cosec x

Question

I don't find an answer for this question matching the solution, please help me out! 

The range is 0 to 2π

 

Solve 2cot²x = 5 - cosec x

Answer 1

I'm going to use trig identities to simplify this. You will want to bone up on the identities. They can save a world of trouble come test-time.

 

cot x = cos x / sin x

csc x = 1 / sin x

 

so 2 cot^2 (x) = 5 - csc (x) means

 

2 cos^2 (x) / sin^2 (x) = 5 - 1 / sin(x)

 

Now, I get rid of the fractions by multiplying all terms by sin^2 (x). I'm going to do that because there are a couple of places in the range of the problem where 1 / sin(x) is undefined. Whenever sin(x) goes to zero.

 

cos^2 (x) = 5 sin^2 (x) - sin (x) = 5 sin(x) (1 - sin(x))

 

This sucker has several solutions in the range, but the explicit solution is hard enough to find that you'll find it a TON easier to solve graphically. plot cos^2 (x) v. x, then plot 5 sin(x) (1 - sin(x)) over the range, and look for where the two plots intersect. You can do this with a graphing calculator. I did it with Wolfram Alpha. And I found the first two solutions at x somewhere around 0.5 and somewhere around 2.7. An explicit expression for the solutions you can get from Wolfram Alpha, but you may need help understanding it. LOL