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

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

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

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

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Factor

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

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

Factor out common term

sin (x)

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

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

Solving each part separately

sin (x) = 0 or 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

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

tan (x) - 3 = 0

Move

3

to the right side

tan (x) - 3 = 0

Add

3

to both sides

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

Simplify

tan (x) = 3

tan (x) = 3

General solutions for

tan (x) = 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}{3} + πn

x = \frac{π}{3} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + πn

cot (\frac{x}{2}) = sin (x)

2 \cdot cos (x) = 3 \cdot tan (x)

cos (θ) = - \frac{24}{25}

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

sin (x) = sec (x)

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