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

The general solution for 2sin(x/3)+sqrt(3)=0 is x=4pi+6pin,x=5pi+6pin

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

Solution

2 sin (\frac{x}{3}) + 3 = 0

Solution

x = 4 π + 6 πn, x = 5 π + 6 πn

Degrees

x = 72 0^{\circ} + 108 0^{\circ} n, x = 90 0^{\circ} + 108 0^{\circ} n

Solution steps

2 sin (\frac{x}{3}) + 3 = 0

Move

3

to the right side

2 sin (\frac{x}{3}) + 3 = 0

Subtract

3

from both sides

2 sin (\frac{x}{3}) + 3 - 3 = 0 - 3

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

\frac{x}{3} = \frac{4 π}{3} + 2 πn, \frac{x}{3} = \frac{5 π}{3} + 2 πn

\frac{x}{3} = \frac{4 π}{3} + 2 πn, \frac{x}{3} = \frac{5 π}{3} + 2 πn

Solve

\frac{x}{3} = \frac{4 π}{3} + 2 πn : x = 4 π + 6 πn

\frac{x}{3} = \frac{4 π}{3} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{4 π}{3} + 2 πn

Multiply both sides by

3

\frac{3 x}{3} = 3 \cdot \frac{4 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} = 3 \cdot \frac{4 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{4 π}{3} + 3 \cdot 2 πn : 4 π + 6 πn

3 \cdot \frac{4 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{4 π}{3} = 4 π

3 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 3}{3}

Cancel the common factor:

3

= 4 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 4 π + 6 πn

x = 4 π + 6 πn

x = 4 π + 6 πn

x = 4 π + 6 πn

Solve

\frac{x}{3} = \frac{5 π}{3} + 2 πn : x = 5 π + 6 πn

\frac{x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{3 x}{3} = 3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} = 3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn : 5 π + 6 πn

3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{3} = 5 π

3 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 3}{3}

Cancel the common factor:

3

= 5 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 5 π + 6 πn

x = 5 π + 6 πn

x = 5 π + 6 πn

x = 5 π + 6 πn

x = 4 π + 6 πn, x = 5 π + 6 πn

Popular Examples

tan(θ)= 15/8

tan (θ) = \frac{15}{8}

sin(θ)=4

sin (θ) = 4

-1=cos(x)+cos(2x)

- 1 = cos (x) + cos (2 x)

16arcsin(x)=4pi

16 arcsin (x) = 4 π

3cos^2(x)-5cos(x)=1+3sin^2(x)

3 cos^{2} (x) - 5 cos (x) = 1 + 3 sin^{2} (x)

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x/3)+sqrt(3)=0 ?
The general solution for 2sin(x/3)+sqrt(3)=0 is x=4pi+6pin,x=5pi+6pin