What is the general solution for 2sin(x)+sqrt(3)tan(x)=0 ?

The general solution for 2sin(x)+sqrt(3)tan(x)=0 is x=2pin,x=pi+2pin,x=(5pi)/6+2pin,x=(7pi)/6+2pin

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2sin(x)+sqrt(3)tan(x)=0

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

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Solution steps

2 sin (x) + 3 tan (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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

tan (t) = 2, 0 < t < \frac{π}{2}, cos (t)

\frac{cos ( x ) + sin ( x )}{sin ( x )} = \frac{1}{tan ( x )}

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

sin (θ) = \frac{1}{4}, 0 < θ < \frac{π}{2}

sin (2 θ) = \frac{1.35 \cdot 10}{( 38 ) ^{2}}

What is the general solution for 2sin(x)+sqrt(3)tan(x)=0 ?
The general solution for 2sin(x)+sqrt(3)tan(x)=0 is x=2pin,x=pi+2pin,x=(5pi)/6+2pin,x=(7pi)/6+2pin