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

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

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Popular Trigonometry >

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

Solution

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

Solution

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

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

3 tan (x)

from both sides

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

Factor

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

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

Factor out common term

tan (x)

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

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

Solving each part separately

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

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

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 = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

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

2 sin (x) - 3 = 0

Move

3

to the right side

2 sin (x) - 3 = 0

Add

3

to both sides

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

Simplify

2 sin (x) = 3

2 sin (x) = 3

Divide both sides by

2

2 sin (x) = 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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

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

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

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