What is the general solution for csc(x)tan(x)=(2sqrt(3))/3 ?

The general solution for csc(x)tan(x)=(2sqrt(3))/3 is x= pi/6+2pin,x=(11pi)/6+2pin

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csc(x)tan(x)=(2sqrt(3))/3

csc (x) tan (x) = \frac{2 3}{3}

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

csc (x) tan (x) = \frac{2 3}{3}

Subtract

\frac{2 3}{3}

from both sides

csc (x) tan (x) - \frac{2}{3} = 0

Simplify

csc (x) tan (x) - \frac{2}{3} : \frac{3 csc ( x ) tan ( x ) - 2}{3}

\frac{3 csc ( x ) tan ( x ) - 2}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 csc (x) tan (x) - 2 = 0

Express with sin, cos

\frac{3 - 2 cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 - 2 cos (x) = 0

Move

3

to the right side

- 2 cos (x) = - 3

Divide both sides by

- 2

cos (x) = \frac{3}{2}

General solutions for

cos (x) = \frac{3}{2}

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 cos^{2} (x) + 2 cos (x) = 1

\frac{- cos ( x ) + sin ( x )}{1 + sin ( 2 x )} = 0

cos^{2} (2 x) - \frac{1}{4} = 0

cosh (x) = \frac{17}{15}

4 cos (x) + 23 = 0

What is the general solution for csc(x)tan(x)=(2sqrt(3))/3 ?
The general solution for csc(x)tan(x)=(2sqrt(3))/3 is x= pi/6+2pin,x=(11pi)/6+2pin