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

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

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

Solution

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

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

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}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \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{a}{- b} = - \frac{a}{b}

= - \frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

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

= - \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= - \frac{π}{6}

= - \frac{π}{6} + \frac{2 πn}{- 2}

\frac{2 πn}{- 2} = - πn

\frac{2 πn}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \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}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \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{a}{- b} = - \frac{a}{b}

= - \frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

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

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

= - \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{π}{3} + \frac{2 πn}{- 2}

\frac{2 πn}{- 2} = - πn

\frac{2 πn}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

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