What is the general solution for 22=18+8cos((x+9)/(12)pi) ?

The general solution for 22=18+8cos((x+9)/(12)pi) is x=24n-5,x=24n+11

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22=18+8cos((x+9)/(12)pi)

Solution

22 = 18 + 8 cos (\frac{x + 9}{12} π)

Solution

x = 24 n - 5, x = 24 n + 11

Degrees

x = - 286.47889 \dots^{\circ} + 1375.09870 \dots^{\circ} n, x = 630.25357 \dots^{\circ} + 1375.09870 \dots^{\circ} n

Solution steps

22 = 18 + 8 cos (\frac{x + 9}{12} π)

Switch sides

18 + 8 cos (\frac{x + 9}{12} π) = 22

Move

18

to the right side

18 + 8 cos (\frac{x + 9}{12} π) = 22

Subtract

18

from both sides

18 + 8 cos (\frac{x + 9}{12} π) - 18 = 22 - 18

Simplify

8 cos (\frac{x + 9}{12} π) = 4

8 cos (\frac{x + 9}{12} π) = 4

Divide both sides by

8

8 cos (\frac{x + 9}{12} π) = 4

Divide both sides by

8

\frac{8 cos ( \frac{x + 9}{12} π )}{8} = \frac{4}{8}

Simplify

cos (\frac{x + 9}{12} π) = \frac{1}{2}

cos (\frac{x + 9}{12} π) = \frac{1}{2}

General solutions for

cos (\frac{x + 9}{12} π) = \frac{1}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{x + 9}{12} π = \frac{π}{3} + 2 πn, \frac{x + 9}{12} π = \frac{5 π}{3} + 2 πn

\frac{x + 9}{12} π = \frac{π}{3} + 2 πn, \frac{x + 9}{12} π = \frac{5 π}{3} + 2 πn

Solve

\frac{x + 9}{12} π = \frac{π}{3} + 2 πn : x = 24 n - 5

\frac{x + 9}{12} π = \frac{π}{3} + 2 πn

Multiply fractions:

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

\frac{π ( x + 9 )}{12} = \frac{π}{3} + 2 πn

Multiply both sides by

12

\frac{π ( x + 9 )}{12} = \frac{π}{3} + 2 πn

Multiply both sides by

12

\frac{12 π ( x + 9 )}{12} = 12 \cdot \frac{π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π ( x + 9 )}{12} = 12 \cdot \frac{π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π ( x + 9 )}{12} : π (x + 9)

\frac{12 π ( x + 9 )}{12}

Divide the numbers:

\frac{12}{12} = 1

= π (x + 9)

Simplify

12 \cdot \frac{π}{3} + 12 \cdot 2 πn : 4 π + 24 πn

12 \cdot \frac{π}{3} + 12 \cdot 2 πn

12 \cdot \frac{π}{3} = 4 π

12 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 12}{3}

Divide the numbers:

\frac{12}{3} = 4

= 4 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 4 π + 24 πn

π (x + 9) = 4 π + 24 πn

π (x + 9) = 4 π + 24 πn

π (x + 9) = 4 π + 24 πn

Divide both sides by

π

π (x + 9) = 4 π + 24 πn

Divide both sides by

π

\frac{π ( x + 9 )}{π} = \frac{4 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π ( x + 9 )}{π} = \frac{4 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π ( x + 9 )}{π} : x + 9

\frac{π ( x + 9 )}{π}

Cancel the common factor:

π

= x + 9

Simplify

\frac{4 π}{π} + \frac{24 πn}{π} : 4 + 24 n

\frac{4 π}{π} + \frac{24 πn}{π}

Cancel

\frac{4 π}{π} : 4

\frac{4 π}{π}

Cancel the common factor:

π

= 4

= 4 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 4 + 24 n

x + 9 = 4 + 24 n

x + 9 = 4 + 24 n

x + 9 = 4 + 24 n

Move

9

to the right side

x + 9 = 4 + 24 n

Subtract

9

from both sides

x + 9 - 9 = 4 + 24 n - 9

Simplify

x = 24 n - 5

x = 24 n - 5

Solve

\frac{x + 9}{12} π = \frac{5 π}{3} + 2 πn : x = 24 n + 11

\frac{x + 9}{12} π = \frac{5 π}{3} + 2 πn

Multiply fractions:

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

\frac{π ( x + 9 )}{12} = \frac{5 π}{3} + 2 πn

Multiply both sides by

12

\frac{π ( x + 9 )}{12} = \frac{5 π}{3} + 2 πn

Multiply both sides by

12

\frac{12 π ( x + 9 )}{12} = 12 \cdot \frac{5 π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π ( x + 9 )}{12} = 12 \cdot \frac{5 π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π ( x + 9 )}{12} : π (x + 9)

\frac{12 π ( x + 9 )}{12}

Divide the numbers:

\frac{12}{12} = 1

= π (x + 9)

Simplify

12 \cdot \frac{5 π}{3} + 12 \cdot 2 πn : 20 π + 24 πn

12 \cdot \frac{5 π}{3} + 12 \cdot 2 πn

12 \cdot \frac{5 π}{3} = 20 π

12 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 12}{3}

Multiply the numbers:

5 \cdot 12 = 60

= \frac{60 π}{3}

Divide the numbers:

\frac{60}{3} = 20

= 20 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 20 π + 24 πn

π (x + 9) = 20 π + 24 πn

π (x + 9) = 20 π + 24 πn

π (x + 9) = 20 π + 24 πn

Divide both sides by

π

π (x + 9) = 20 π + 24 πn

Divide both sides by

π

\frac{π ( x + 9 )}{π} = \frac{20 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π ( x + 9 )}{π} = \frac{20 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π ( x + 9 )}{π} : x + 9

\frac{π ( x + 9 )}{π}

Cancel the common factor:

π

= x + 9

Simplify

\frac{20 π}{π} + \frac{24 πn}{π} : 20 + 24 n

\frac{20 π}{π} + \frac{24 πn}{π}

Cancel

\frac{20 π}{π} : 20

\frac{20 π}{π}

Cancel the common factor:

π

= 20

= 20 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 20 + 24 n

x + 9 = 20 + 24 n

x + 9 = 20 + 24 n

x + 9 = 20 + 24 n

Move

9

to the right side

x + 9 = 20 + 24 n

Subtract

9

from both sides

x + 9 - 9 = 20 + 24 n - 9

Simplify

x = 24 n + 11

x = 24 n + 11

x = 24 n - 5, x = 24 n + 11

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Frequently Asked Questions (FAQ)

What is the general solution for 22=18+8cos((x+9)/(12)pi) ?
The general solution for 22=18+8cos((x+9)/(12)pi) is x=24n-5,x=24n+11