What is the general solution for cos(1/(3x))= 1/3 ?

The general solution for cos(1/(3x))= 1/3 is x= 1/(3(1.23095…+2pin)),x= 1/(3(2pi-1.23095…+2pin))

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cos(1/(3x))= 1/3

Solution

cos (\frac{1}{3 x}) = \frac{1}{3}

Solution

x = \frac{1}{3 ( 1.23095 \dots + 2 πn )}, x = \frac{1}{3 ( 2 π - 1.23095 \dots + 2 πn )}

Degrees

x = 0^{\circ} + 2.54168 \dots^{\circ} n, x = 0^{\circ} + 1.68486 \dots^{\circ} n

Solution steps

cos (\frac{1}{3 x}) = \frac{1}{3}

Apply trig inverse properties

cos (\frac{1}{3 x}) = \frac{1}{3}

General solutions for

cos (\frac{1}{3 x}) = \frac{1}{3}

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

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

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

Solve

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

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

Multiply both sides by

x

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

Multiply both sides by

x

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

Simplify

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

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

Switch sides

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

Factor

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

arccos (\frac{1}{3}) x + 2 πn x

Factor out common term

x

= x (arccos (\frac{1}{3}) + 2 πn)

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

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) = \frac{1}{3}

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

Cancel the common factor:

arccos (\frac{1}{3}) + 2 πn

= x

Simplify

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

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

Apply the fraction rule:

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

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

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

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

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

Solve

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

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

Multiply both sides by

x

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

Multiply both sides by

x

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

Simplify

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

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

Switch sides

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

Factor

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

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

Factor out common term

x

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

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

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) = \frac{1}{3}

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

Cancel the common factor:

2 π - arccos (\frac{1}{3}) + 2 πn

= x

Simplify

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

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

Apply the fraction rule:

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

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

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

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

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

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

Show solutions in decimal form

x = \frac{1}{3 ( 1.23095 \dots + 2 πn )}, x = \frac{1}{3 ( 2 π - 1.23095 \dots + 2 πn )}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(1/(3x))= 1/3 ?
The general solution for cos(1/(3x))= 1/3 is x= 1/(3(1.23095…+2pin)),x= 1/(3(2pi-1.23095…+2pin))