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

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

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

Solution

2 sin (\frac{x}{2} + \frac{π}{3}) = 3

Solution

x = 4 πn, x = 4 πn + \frac{2 π}{3}

Degrees

x = 0^{\circ} + 72 0^{\circ} n, x = 12 0^{\circ} + 72 0^{\circ} n

Solution steps

2 sin (\frac{x}{2} + \frac{π}{3}) = 3

Divide both sides by

2

2 sin (\frac{x}{2} + \frac{π}{3}) = 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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

Solve

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

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

Subtract

\frac{π}{3}

from both sides

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

Simplify

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplify

x = 4 πn

x = 4 πn

Solve

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

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

Move

\frac{π}{3}

to the right side

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

Subtract

\frac{π}{3}

from both sides

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

Simplify

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

Simplify

\frac{x}{2} + \frac{π}{3} - \frac{π}{3} : \frac{x}{2}

\frac{x}{2} + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= \frac{x}{2}

Simplify

\frac{2 π}{3} + 2 πn - \frac{π}{3} : 2 πn + \frac{π}{3}

\frac{2 π}{3} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn - \frac{π}{3} + \frac{2 π}{3}

Combine the fractions

- \frac{π}{3} + \frac{2 π}{3} : \frac{π}{3}

Apply rule

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

= \frac{- π + 2 π}{3}

Add similar elements:

- π + 2 π = π

= \frac{π}{3}

= 2 πn + \frac{π}{3}

\frac{x}{2} = 2 πn + \frac{π}{3}

\frac{x}{2} = 2 πn + \frac{π}{3}

\frac{x}{2} = 2 πn + \frac{π}{3}

Multiply both sides by

2

\frac{x}{2} = 2 πn + \frac{π}{3}

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot 2 πn + 2 \cdot \frac{π}{3}

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 2 \cdot \frac{π}{3}

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= 4 πn + \frac{2 π}{3}

x = 4 πn + \frac{2 π}{3}

x = 4 πn + \frac{2 π}{3}

x = 4 πn + \frac{2 π}{3}

x = 4 πn, x = 4 πn + \frac{2 π}{3}

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Frequently Asked Questions (FAQ)

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