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

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

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

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

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

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Factor

3 tan (x) sin (x) - 3 sin (x) : sin (x) (3 tan (x) - 3)

3 tan (x) sin (x) - 3 sin (x)

Factor out common term

sin (x)

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

3 tan (x) - 3 = 0 : x = \frac{π}{6} + πn

3 tan (x) - 3 = 0

Move

3

to the right side

3 tan (x) - 3 = 0

Add

3

to both sides

3 tan (x) - 3 + 3 = 0 + 3

Simplify

3 tan (x) = 3

3 tan (x) = 3

Divide both sides by

3

3 tan (x) = 3

Divide both sides by

3

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

Simplify

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

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

General solutions for

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

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

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

cot^{4} (x) - 3 cot^{2} (x) + 1 = 0

\frac{cos ( - 3 0 ^{\circ} + x )}{cos ( 3 0 ^{\circ} + x )} = \frac{1835}{726}

sin^{2} (x) = 3 cos (x) - 2

4 sin (2 x + \frac{π}{6}) = 2

8 cos (θ) = 3 - 4 cos (θ)

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