1. Trigonometric equation

how can i solve this equation 2cot^2(2x)+3cosec2x=0 for o<x<π??

2. Re: Trigonometric equation

Endri
Substitute cot(2x) by cosx/sinx and cosec2x by 1/sin2x and do some work to get a quadratic then solve it it is easy.
good luck

3. Re: Trigonometric equation

Hello, endri!

$\text{Solve: }\:2\cot^2(2x)+3\csc(2x)\:=\:0\,\text{ for }0 < x < \pi.$

We have: . $2\cot^2(2x) + 3\csc(2x) \:=\:0$

. . . . $2[\csc^2(2x) - 1] + 3\csc(2x) \:=\:0$

. . . . . $2\csc^2(2x) - 2 + 3\csc(2x) \:=\:0$

. . . . . $2\csc^2(2x) + 3\csc(2x) - 2 \:=\:0$

. . . . $[2\csc(2x) - 1][\csc(2x)+2] \:=\:0$

Then: . $\begin{Bmatrix}2\csc(2x) - 1 \:=\:0 & \Rightarrow & \csc(2x) \:=\:\frac{1}{2} && \text{No real roots} \\ \csc(2x) + 2\:=\:0 & \Rightarrow & \csc(2x) \:=\:-2 & \Rightarrow & 2x \:=\:\frac{7\pi}{6},\,\frac{11\pi}{6} & \Rightarrow & \boxed{x \:=\:\tfrac{7\pi}{12},\,\tfrac{11\pi}{12}} \end{Bmatrix}$

4. Re: Trigonometric equation

I strongly believe that we should guide the readers to do some work by themselves and study more rather than solve them their homework.

MINOAS

5. Re: Trigonometric equation

THANK YOU VERY MUCH MAN

6. Re: Trigonometric equation

cosec^2(x)=1+cot^2(x),,and i wasnt sure if it was right to say that cosec^2(2x)=1+cot^2(2x)... thanks again man

7. Re: Trigonometric equation

minoanaman thanks for your help too man.. i had solved this equation,but i was not sure if the method that soroban showed me was right