What is the general solution for cos(pi/x)=(sqrt(3))/3 ?

The general solution for cos(pi/x)=(sqrt(3))/3 is x= pi/(0.95531…+2pin),x= pi/(2pi-0.95531…+2pin)

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cos(pi/x)=(sqrt(3))/3

Solution

cos (\frac{π}{x}) = \frac{3}{3}

Solution

x = \frac{π}{0.95531 \dots + 2 πn}, x = \frac{π}{2 π - 0.95531 \dots + 2 πn}

Degrees

x = 0^{\circ} + 24.86702 \dots^{\circ} n, x = 0^{\circ} + 15.50246 \dots^{\circ} n

Solution steps

cos (\frac{π}{x}) = \frac{3}{3}

Apply trig inverse properties

cos (\frac{π}{x}) = \frac{3}{3}

General solutions for

cos (\frac{π}{x}) = \frac{3}{3}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

Solve

\frac{π}{x} = arccos (\frac{3}{3}) + 2 πn : x = \frac{π}{arccos ( \frac{1}{3} ) + 2 πn}; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

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

Multiply both sides by

x

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

Multiply both sides by

x

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

Simplify

π = arccos (\frac{3}{3}) x + 2 πn x

π = arccos (\frac{3}{3}) x + 2 πn x

Switch sides

arccos (\frac{3}{3}) x + 2 πn x = π

Simplify

arccos (\frac{3}{3}) x + 2 πn x : arccos (\frac{1}{3}) x + 2 πn x

arccos (\frac{3}{3}) x + 2 πn x

\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}

= arccos (\frac{1}{3}) x + 2 πn x

arccos (\frac{1}{3}) x + 2 πn x = π

arccos (\frac{3}{3}) x + 2 πn x = π

Factor

arccos (\frac{3}{3}) x + 2 πn x : x (arccos (\frac{1}{3}) + 2 πn)

arccos (\frac{3}{3}) x + 2 πn x

Factor out common term

x

= x (arccos (\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}

= x (2 πn + arccos (\frac{1}{3}))

x (arccos (\frac{1}{3}) + 2 πn) = π

Divide both sides by

arccos (\frac{1}{3}) + 2 πn; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

x (arccos (\frac{1}{3}) + 2 πn) = π

Divide both sides by

arccos (\frac{1}{3}) + 2 πn; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

\frac{x ( arccos ( \frac{1}{3} ) + 2 πn )}{arccos ( \frac{1}{3} ) + 2 πn} = \frac{π}{arccos ( \frac{1}{3} ) + 2 πn}; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

Simplify

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

Simplify

\frac{x ( arccos ( \frac{1}{3} ) + 2 πn )}{arccos ( \frac{1}{3} ) + 2 πn} : x

\frac{x ( arccos ( \frac{1}{3} ) + 2 πn )}{arccos ( \frac{1}{3} ) + 2 πn}

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

x (arccos (\frac{1}{3}) + 2 πn)

= x (2 πn + arccos (\frac{3}{3}))

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

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

arccos (\frac{1}{3}) + 2 πn

= arccos (\frac{3}{3}) + 2 πn

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

Cancel the common factor:

arccos (\frac{3}{3}) + 2 πn

= x

Simplify

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

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

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

arccos (\frac{1}{3}) + 2 πn

= arccos (\frac{3}{3}) + 2 πn

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

x = \frac{π}{arccos ( \frac{3}{3} ) + 2 πn}; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

x = \frac{π}{arccos ( \frac{3}{3} ) + 2 πn}; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

x = \frac{π}{arccos ( \frac{3}{3} ) + 2 πn}; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

Simplify

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

\frac{π}{arccos ( \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}

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

x = \frac{π}{arccos ( \frac{1}{3} ) + 2 πn}; n \neq = - \frac{arccos ( \frac{1}{3} )}{2 π}

Solve

\frac{π}{x} = 2 π - arccos (\frac{3}{3}) + 2 πn : x = \frac{π}{2 π - arccos ( \frac{1}{3} ) + 2 πn}; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

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

Multiply both sides by

x

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

Multiply both sides by

x

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

Simplify

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

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

Switch sides

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

Simplify

2 π x - arccos (\frac{3}{3}) x + 2 πn x : 2 π x - arccos (\frac{1}{3}) x + 2 πn x

2 π x - arccos (\frac{3}{3}) x + 2 πn x

\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}

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

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

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

Factor

2 π x - arccos (\frac{3}{3}) x + 2 πn x : x (2 π - arccos (\frac{1}{3}) + 2 πn)

2 π x - arccos (\frac{3}{3}) x + 2 πn x

Factor out common term

x

= x (2 π - arccos (\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}

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

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

Divide both sides by

2 π - arccos (\frac{1}{3}) + 2 πn; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

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

Divide both sides by

2 π - arccos (\frac{1}{3}) + 2 πn; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

\frac{x ( 2 π - arccos ( \frac{1}{3} ) + 2 πn )}{2 π - arccos ( \frac{1}{3} ) + 2 πn} = \frac{π}{2 π - arccos ( \frac{1}{3} ) + 2 πn}; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

Simplify

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

Simplify

\frac{x ( 2 π - arccos ( \frac{1}{3} ) + 2 πn )}{2 π - arccos ( \frac{1}{3} ) + 2 πn} : x

\frac{x ( 2 π - arccos ( \frac{1}{3} ) + 2 πn )}{2 π - arccos ( \frac{1}{3} ) + 2 πn}

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

x (2 π - arccos (\frac{1}{3}) + 2 πn)

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

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

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

2 π - arccos (\frac{1}{3}) + 2 πn

= 2 π - arccos (\frac{3}{3}) + 2 πn

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

Cancel the common factor:

2 π - arccos (\frac{3}{3}) + 2 πn

= x

Simplify

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

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

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

2 π - arccos (\frac{1}{3}) + 2 πn

= 2 π - arccos (\frac{3}{3}) + 2 πn

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

x = \frac{π}{2 π - arccos ( \frac{3}{3} ) + 2 πn}; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

x = \frac{π}{2 π - arccos ( \frac{3}{3} ) + 2 πn}; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

x = \frac{π}{2 π - arccos ( \frac{3}{3} ) + 2 πn}; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

Simplify

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

\frac{π}{2 π - arccos ( \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}

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

x = \frac{π}{2 π - arccos ( \frac{1}{3} ) + 2 πn}; n \neq = \frac{- 2 π + arccos ( \frac{1}{3} )}{2 π}

x = \frac{π}{arccos ( \frac{1}{3} ) + 2 πn}, x = \frac{π}{2 π - arccos ( \frac{1}{3} ) + 2 πn}

Show solutions in decimal form

x = \frac{π}{0.95531 \dots + 2 πn}, x = \frac{π}{2 π - 0.95531 \dots + 2 πn}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(pi/x)=(sqrt(3))/3 ?
The general solution for cos(pi/x)=(sqrt(3))/3 is x= pi/(0.95531…+2pin),x= pi/(2pi-0.95531…+2pin)