What is the general solution for 1/3 =cos((pix)/(12)) ?

The general solution for 1/3 =cos((pix)/(12)) is x=(12*1.23095…)/pi+24n,x=24-(12*1.23095…)/pi+24n

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1/3 =cos((pix)/(12))

Solution

\frac{1}{3} = cos (\frac{π x}{12})

Solution

x = \frac{12 \cdot 1.23095 \dots}{π} + 24 n, x = 24 - \frac{12 \cdot 1.23095 \dots}{π} + 24 n

Degrees

x = 269.40009 \dots^{\circ} + 1375.09870 \dots^{\circ} n, x = 1105.69861 \dots^{\circ} + 1375.09870 \dots^{\circ} n

Solution steps

\frac{1}{3} = cos (\frac{π x}{12})

Switch sides

cos (\frac{π x}{12}) = \frac{1}{3}

Apply trig inverse properties

cos (\frac{π x}{12}) = \frac{1}{3}

General solutions for

cos (\frac{π x}{12}) = \frac{1}{3}

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

\frac{π x}{12} = arccos (\frac{1}{3}) + 2 πn, \frac{π x}{12} = 2 π - arccos (\frac{1}{3}) + 2 πn

\frac{π x}{12} = arccos (\frac{1}{3}) + 2 πn, \frac{π x}{12} = 2 π - arccos (\frac{1}{3}) + 2 πn

Solve

\frac{π x}{12} = arccos (\frac{1}{3}) + 2 πn : x = \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

\frac{π x}{12} = arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

12

\frac{π x}{12} = arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

12

\frac{12 π x}{12} = 12 arccos (\frac{1}{3}) + 12 \cdot 2 πn

Simplify

\frac{12 π x}{12} = 12 arccos (\frac{1}{3}) + 12 \cdot 2 πn

Simplify

\frac{12 π x}{12} : π x

\frac{12 π x}{12}

Divide the numbers:

\frac{12}{12} = 1

= π x

Simplify

12 arccos (\frac{1}{3}) + 12 \cdot 2 πn : 12 arccos (\frac{1}{3}) + 24 πn

12 arccos (\frac{1}{3}) + 12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 12 arccos (\frac{1}{3}) + 24 πn

π x = 12 arccos (\frac{1}{3}) + 24 πn

π x = 12 arccos (\frac{1}{3}) + 24 πn

π x = 12 arccos (\frac{1}{3}) + 24 πn

Divide both sides by

π

π x = 12 arccos (\frac{1}{3}) + 24 πn

Divide both sides by

π

\frac{π x}{π} = \frac{12 arccos ( \frac{1}{3} )}{π} + \frac{24 πn}{π}

Simplify

x = \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

x = \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

Solve

\frac{π x}{12} = 2 π - arccos (\frac{1}{3}) + 2 πn : x = 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

\frac{π x}{12} = 2 π - arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

12

\frac{π x}{12} = 2 π - arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

12

\frac{12 π x}{12} = 12 \cdot 2 π - 12 arccos (\frac{1}{3}) + 12 \cdot 2 πn

Simplify

\frac{12 π x}{12} = 12 \cdot 2 π - 12 arccos (\frac{1}{3}) + 12 \cdot 2 πn

Simplify

\frac{12 π x}{12} : π x

\frac{12 π x}{12}

Divide the numbers:

\frac{12}{12} = 1

= π x

Simplify

12 \cdot 2 π - 12 arccos (\frac{1}{3}) + 12 \cdot 2 πn : 24 π - 12 arccos (\frac{1}{3}) + 24 πn

12 \cdot 2 π - 12 arccos (\frac{1}{3}) + 12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 π - 12 arccos (\frac{1}{3}) + 24 πn

π x = 24 π - 12 arccos (\frac{1}{3}) + 24 πn

π x = 24 π - 12 arccos (\frac{1}{3}) + 24 πn

π x = 24 π - 12 arccos (\frac{1}{3}) + 24 πn

Divide both sides by

π

π x = 24 π - 12 arccos (\frac{1}{3}) + 24 πn

Divide both sides by

π

\frac{π x}{π} = \frac{24 π}{π} - \frac{12 arccos ( \frac{1}{3} )}{π} + \frac{24 πn}{π}

Simplify

\frac{π x}{π} = \frac{24 π}{π} - \frac{12 arccos ( \frac{1}{3} )}{π} + \frac{24 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{24 π}{π} - \frac{12 arccos ( \frac{1}{3} )}{π} + \frac{24 πn}{π} : 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

\frac{24 π}{π} - \frac{12 arccos ( \frac{1}{3} )}{π} + \frac{24 πn}{π}

Cancel

\frac{24 π}{π} : 24

\frac{24 π}{π}

Cancel the common factor:

π

= 24

= 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

x = 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

x = 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

x = 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

x = \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n, x = 24 - \frac{12 arccos ( \frac{1}{3} )}{π} + 24 n

Show solutions in decimal form

x = \frac{12 \cdot 1.23095 \dots}{π} + 24 n, x = 24 - \frac{12 \cdot 1.23095 \dots}{π} + 24 n

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Frequently Asked Questions (FAQ)

What is the general solution for 1/3 =cos((pix)/(12)) ?
The general solution for 1/3 =cos((pix)/(12)) is x=(12*1.23095…)/pi+24n,x=24-(12*1.23095…)/pi+24n