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

The general solution for sqrt(3)tan(x)-2sin(x)=0 is x=2pin,x=pi+2pin,x= pi/6+2pin,x=(11pi)/6+2pin

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

Solution

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

Solution

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

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

- 2 sin (x) + 3 \frac{sin ( x )}{cos ( x )} : \frac{- 2 sin ( x ) cos ( x ) + 3 sin ( x )}{cos ( x )}

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

Multiply

3 \frac{sin ( x )}{cos ( x )} : \frac{3 sin ( x )}{cos ( x )}

3 \frac{sin ( x )}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

\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) sin (x)

Factor out common term

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

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

3 - 2 cos (x) = 0

Move

3

to the right side

3 - 2 cos (x) = 0

Subtract

3

from both sides

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

Simplify

- 2 cos (x) = - 3

- 2 cos (x) = - 3

Divide both sides by

- 2

- 2 cos (x) = - 3

Divide both sides by

- 2

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

Simplify

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

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Popular Examples

2sin^2(2x)-cos(2x)-1=0

2 sin^{2} (2 x) - cos (2 x) - 1 = 0

2*cos(x)-1=0

2 \cdot cos (x) - 1 = 0

cos(x)+sqrt(3)sin(x)=2

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

2sin(θ)=-sqrt(2),0<= θ<2pi

2 sin (θ) = - 2, 0 \leq θ < 2 π

solvefor y,x=arcsin(y^2)solve for

y, x = arcsin (y^{2})

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)tan(x)-2sin(x)=0 ?
The general solution for sqrt(3)tan(x)-2sin(x)=0 is x=2pin,x=pi+2pin,x= pi/6+2pin,x=(11pi)/6+2pin