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

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

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

Solution

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

Solution

x = \frac{π}{6} + πn, x = \frac{π}{3} + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Multiply

\frac{3}{2} 3 : \frac{3 3}{2}

\frac{3}{2} 3

Multiply fractions:

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

= \frac{3 3}{2}

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

Move

\frac{3 3}{2}

to the right side

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

Add

\frac{3 3}{2}

to both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

Simplify

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

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

Divide the numbers:

\frac{3}{3} = 1

= sin (2 x)

Simplify

\frac{\frac{3 3}{2}}{3} : \frac{3}{2}

\frac{\frac{3 3}{2}}{3}

Apply the fraction rule:

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

= \frac{3 3}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 3}{6}

Cancel the common factor:

3

= \frac{3}{2}

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

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

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

General solutions for

sin (2 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}

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

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

Solve

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

2 x = \frac{π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Solve

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

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

2 x = \frac{2 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{6} + πn, x = \frac{π}{3} + πn

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Frequently Asked Questions (FAQ)

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