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

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

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sqrt(3)tan(x)=2sin(x)

3 tan (x) = 2 sin (x)

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

Solution steps

3 tan (x) = 2 sin (x)

Subtract

2 sin (x)

from both sides

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

Express with sin, cos

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

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

sin (x) 3 - 2 cos (x) sin (x) = 0

Factor

sin (x) 3 - 2 cos (x) sin (x) : sin (x) (3 - 2 cos (x))

sin (x) (3 - 2 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 3 - 2 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

3 - 2 cos (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

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

12 arcsin (x) = 4 π

sec (x) = 2, 0 \leq x \leq 2 π

cos (2 x) + cos (3 x) = 0

cos (θ) = - 0.73

csc (θ) = - 3

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