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

The general solution for 2sin(x/2)+sqrt(3)=0 is x=(8pi)/3+4pin,x=(10pi)/3+4pin

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

Solution

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

Solution

x = \frac{8 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn

Degrees

x = 48 0^{\circ} + 72 0^{\circ} n, x = 60 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Move

3

to the right side

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

Subtract

3

from both sides

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

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

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

General solutions for

sin (\frac{x}{2}) = - \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}{2} = \frac{4 π}{3} + 2 πn, \frac{x}{2} = \frac{5 π}{3} + 2 πn

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

Solve

\frac{x}{2} = \frac{4 π}{3} + 2 πn : x = \frac{8 π}{3} + 4 πn

\frac{x}{2} = \frac{4 π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{4 π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn : \frac{8 π}{3} + 4 πn

2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{4 π}{3} = \frac{8 π}{3}

2 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 2}{3}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{8 π}{3} + 4 πn

x = \frac{8 π}{3} + 4 πn

x = \frac{8 π}{3} + 4 πn

x = \frac{8 π}{3} + 4 πn

Solve

\frac{x}{2} = \frac{5 π}{3} + 2 πn : x = \frac{10 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn : \frac{10 π}{3} + 4 πn

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{3} = \frac{10 π}{3}

2 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 2}{3}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{8 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn

Popular Examples

2cos^2(x)+3sin(x)=0

2 cos^{2} (x) + 3 sin (x) = 0

tan(3x)-1=0

tan (3 x) - 1 = 0

2sin(2x)-1=0

2 sin (2 x) - 1 = 0

sin(2x)=(sqrt(2))/2

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

csc(x)+1=0

csc (x) + 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x/2)+sqrt(3)=0 ?
The general solution for 2sin(x/2)+sqrt(3)=0 is x=(8pi)/3+4pin,x=(10pi)/3+4pin