What is the general solution for sin(2x)=(-1)/(sqrt(3)) ?

The general solution for sin(2x)=(-1)/(sqrt(3)) is x=-(0.61547…)/2+pin,x= pi/2+(0.61547…)/2+pin

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sin(2x)=(-1)/(sqrt(3))

Solution

sin (2 x) = \frac{- 1}{3}

Solution

x = - \frac{0.61547 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.61547 \dots}{2} + πn

Radians

x = - 0.30773 \dots + πn, x = \frac{π}{2} + \frac{0.61547 \dots}{2} + πn

Solution steps

sin (2 x) = \frac{- 1}{3}

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

2 x = arcsin (- \frac{3}{3}) + 2 πn, 2 x = π + arcsin (\frac{3}{3}) + 2 πn

2 x = arcsin (- \frac{3}{3}) + 2 πn, 2 x = π + arcsin (\frac{3}{3}) + 2 πn

Solve

2 x = arcsin (- \frac{3}{3}) + 2 πn : x = - \frac{arcsin ( \frac{3}{3} )}{2} + πn

2 x = arcsin (- \frac{3}{3}) + 2 πn

Simplify

arcsin (- \frac{3}{3}) + 2 πn : - arcsin (\frac{1}{3}) + 2 πn

arcsin (- \frac{3}{3}) + 2 πn

arcsin (- \frac{3}{3}) = - arcsin (\frac{3}{3})

arcsin (- \frac{3}{3})

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}}}{3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{1}{2}}}{3 ^{1}} = \frac{1}{3 ^{1 - \frac{1}{2}}}

= \frac{1}{3 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{1}{3}

= arcsin (- \frac{1}{3})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - arcsin (\frac{3}{3})

= - arcsin (\frac{3}{3}) + 2 πn

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}}}{3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{1}{2}}}{3 ^{1}} = \frac{1}{3 ^{1 - \frac{1}{2}}}

= \frac{1}{3 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{1}{3}

= - arcsin (\frac{1}{3}) + 2 πn

2 x = - arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

2 x = - arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{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{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} : - \frac{arcsin ( \frac{3}{3} )}{2} + πn

- \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{arcsin ( \frac{1}{3} )}{2} + πn

arcsin (\frac{1}{3}) = arcsin (\frac{3}{3})

arcsin (\frac{1}{3})

= arcsin (\frac{3}{3})

= - \frac{arcsin ( \frac{3}{3} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{3} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{3} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{3} )}{2} + πn

Solve

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

2 x = π + arcsin (\frac{3}{3}) + 2 πn

Simplify

π + arcsin (\frac{3}{3}) + 2 πn : π + arcsin (\frac{1}{3}) + 2 πn

π + arcsin (\frac{3}{3}) + 2 πn

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}}}{3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{1}{2}}}{3 ^{1}} = \frac{1}{3 ^{1 - \frac{1}{2}}}

= \frac{1}{3 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{1}{3}

= π + arcsin (\frac{1}{3}) + 2 πn

2 x = π + arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

2 x = π + arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{1}{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{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} : \frac{π}{2} + \frac{arcsin ( \frac{3}{3} )}{2} + πn

\frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + πn

arcsin (\frac{1}{3}) = arcsin (\frac{3}{3})

arcsin (\frac{1}{3})

= arcsin (\frac{3}{3})

= \frac{π}{2} + \frac{arcsin ( \frac{3}{3} )}{2} + πn

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

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

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

x = - \frac{arcsin ( \frac{3}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{3} )}{2} + πn

Show solutions in decimal form

x = - \frac{0.61547 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.61547 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=(-1)/(sqrt(3)) ?
The general solution for sin(2x)=(-1)/(sqrt(3)) is x=-(0.61547…)/2+pin,x= pi/2+(0.61547…)/2+pin