What is the general solution for 40+40cos((2pix)/9)=50 ?

The general solution for 40+40cos((2pix)/9)=50 is x=(9*1.31811…)/(2pi)+9n,x=9-(9*1.31811…)/(2pi)+9n

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40+40cos((2pix)/9)=50

Solution

40 + 40 cos (\frac{2 π x}{9}) = 50

Solution

x = \frac{9 \cdot 1.31811 \dots}{2 π} + 9 n, x = 9 - \frac{9 \cdot 1.31811 \dots}{2 π} + 9 n

Degrees

x = 108.17799 \dots^{\circ} + 515.66201 \dots^{\circ} n, x = 407.48402 \dots^{\circ} + 515.66201 \dots^{\circ} n

Solution steps

40 + 40 cos (\frac{2 π x}{9}) = 50

Move

40

to the right side

40 + 40 cos (\frac{2 π x}{9}) = 50

Subtract

40

from both sides

40 + 40 cos (\frac{2 π x}{9}) - 40 = 50 - 40

Simplify

40 cos (\frac{2 π x}{9}) = 10

40 cos (\frac{2 π x}{9}) = 10

Divide both sides by

40

40 cos (\frac{2 π x}{9}) = 10

Divide both sides by

40

\frac{40 cos ( \frac{2 π x}{9} )}{40} = \frac{10}{40}

Simplify

cos (\frac{2 π x}{9}) = \frac{1}{4}

cos (\frac{2 π x}{9}) = \frac{1}{4}

Apply trig inverse properties

cos (\frac{2 π x}{9}) = \frac{1}{4}

General solutions for

cos (\frac{2 π x}{9}) = \frac{1}{4}

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

\frac{2 π x}{9} = arccos (\frac{1}{4}) + 2 πn, \frac{2 π x}{9} = 2 π - arccos (\frac{1}{4}) + 2 πn

\frac{2 π x}{9} = arccos (\frac{1}{4}) + 2 πn, \frac{2 π x}{9} = 2 π - arccos (\frac{1}{4}) + 2 πn

Solve

\frac{2 π x}{9} = arccos (\frac{1}{4}) + 2 πn : x = \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

\frac{2 π x}{9} = arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

9

\frac{2 π x}{9} = arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

9

\frac{9 \cdot 2 π x}{9} = 9 arccos (\frac{1}{4}) + 9 \cdot 2 πn

Simplify

\frac{9 \cdot 2 π x}{9} = 9 arccos (\frac{1}{4}) + 9 \cdot 2 πn

Simplify

\frac{9 \cdot 2 π x}{9} : 2 π x

\frac{9 \cdot 2 π x}{9}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{18 π x}{9}

Divide the numbers:

\frac{18}{9} = 2

= 2 π x

Simplify

9 arccos (\frac{1}{4}) + 9 \cdot 2 πn : 9 arccos (\frac{1}{4}) + 18 πn

9 arccos (\frac{1}{4}) + 9 \cdot 2 πn

Multiply the numbers:

9 \cdot 2 = 18

= 9 arccos (\frac{1}{4}) + 18 πn

2 π x = 9 arccos (\frac{1}{4}) + 18 πn

2 π x = 9 arccos (\frac{1}{4}) + 18 πn

2 π x = 9 arccos (\frac{1}{4}) + 18 πn

Divide both sides by

2 π

2 π x = 9 arccos (\frac{1}{4}) + 18 πn

Divide both sides by

2 π

\frac{2 π x}{2 π} = \frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π}

Simplify

\frac{2 π x}{2 π} = \frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π}

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π} : \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

\frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π}

Cancel

\frac{18 πn}{2 π} : 9 n

\frac{18 πn}{2 π}

Cancel

\frac{18 πn}{2 π} : 9 n

\frac{18 πn}{2 π}

Divide the numbers:

\frac{18}{2} = 9

= \frac{9 πn}{π}

Cancel the common factor:

π

= 9 n

= 9 n

= \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

x = \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

x = \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

x = \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

Solve

\frac{2 π x}{9} = 2 π - arccos (\frac{1}{4}) + 2 πn : x = 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

\frac{2 π x}{9} = 2 π - arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

9

\frac{2 π x}{9} = 2 π - arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

9

\frac{9 \cdot 2 π x}{9} = 9 \cdot 2 π - 9 arccos (\frac{1}{4}) + 9 \cdot 2 πn

Simplify

\frac{9 \cdot 2 π x}{9} = 9 \cdot 2 π - 9 arccos (\frac{1}{4}) + 9 \cdot 2 πn

Simplify

\frac{9 \cdot 2 π x}{9} : 2 π x

\frac{9 \cdot 2 π x}{9}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{18 π x}{9}

Divide the numbers:

\frac{18}{9} = 2

= 2 π x

Simplify

9 \cdot 2 π - 9 arccos (\frac{1}{4}) + 9 \cdot 2 πn : 18 π - 9 arccos (\frac{1}{4}) + 18 πn

9 \cdot 2 π - 9 arccos (\frac{1}{4}) + 9 \cdot 2 πn

Multiply the numbers:

9 \cdot 2 = 18

= 18 π - 9 arccos (\frac{1}{4}) + 18 πn

2 π x = 18 π - 9 arccos (\frac{1}{4}) + 18 πn

2 π x = 18 π - 9 arccos (\frac{1}{4}) + 18 πn

2 π x = 18 π - 9 arccos (\frac{1}{4}) + 18 πn

Divide both sides by

2 π

2 π x = 18 π - 9 arccos (\frac{1}{4}) + 18 πn

Divide both sides by

2 π

\frac{2 π x}{2 π} = \frac{18 π}{2 π} - \frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π}

Simplify

\frac{2 π x}{2 π} = \frac{18 π}{2 π} - \frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π}

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{18 π}{2 π} - \frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π} : 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

\frac{18 π}{2 π} - \frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π}

Cancel

\frac{18 π}{2 π} : 9

\frac{18 π}{2 π}

Cancel

\frac{18 π}{2 π} : 9

\frac{18 π}{2 π}

Divide the numbers:

\frac{18}{2} = 9

= \frac{9 π}{π}

Cancel the common factor:

π

= 9

= 9

= 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + \frac{18 πn}{2 π}

Cancel

\frac{18 πn}{2 π} : 9 n

\frac{18 πn}{2 π}

Cancel

\frac{18 πn}{2 π} : 9 n

\frac{18 πn}{2 π}

Divide the numbers:

\frac{18}{2} = 9

= \frac{9 πn}{π}

Cancel the common factor:

π

= 9 n

= 9 n

= 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

x = 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

x = 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

x = 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

x = \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n, x = 9 - \frac{9 arccos ( \frac{1}{4} )}{2 π} + 9 n

Show solutions in decimal form

x = \frac{9 \cdot 1.31811 \dots}{2 π} + 9 n, x = 9 - \frac{9 \cdot 1.31811 \dots}{2 π} + 9 n

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Frequently Asked Questions (FAQ)

What is the general solution for 40+40cos((2pix)/9)=50 ?
The general solution for 40+40cos((2pix)/9)=50 is x=(9*1.31811…)/(2pi)+9n,x=9-(9*1.31811…)/(2pi)+9n