What is the general solution for 300=400+200cos((pit)/(12)) ?

The general solution for 300=400+200cos((pit)/(12)) is t=8+24n,t=16+24n

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300=400+200cos((pit)/(12))

Solution

300 = 400 + 200 cos (\frac{π t}{12})

Solution

t = 8 + 24 n, t = 16 + 24 n

Degrees

t = 458.36623 \dots^{\circ} + 1375.09870 \dots^{\circ} n, t = 916.73247 \dots^{\circ} + 1375.09870 \dots^{\circ} n

Solution steps

300 = 400 + 200 cos (\frac{π t}{12})

Switch sides

400 + 200 cos (\frac{π t}{12}) = 300

Move

400

to the right side

400 + 200 cos (\frac{π t}{12}) = 300

Subtract

400

from both sides

400 + 200 cos (\frac{π t}{12}) - 400 = 300 - 400

Simplify

200 cos (\frac{π t}{12}) = - 100

200 cos (\frac{π t}{12}) = - 100

Divide both sides by

200

200 cos (\frac{π t}{12}) = - 100

Divide both sides by

200

\frac{200 cos ( \frac{π t}{12} )}{200} = \frac{- 100}{200}

Simplify

\frac{200 cos ( \frac{π t}{12} )}{200} = \frac{- 100}{200}

Simplify

\frac{200 cos ( \frac{π t}{12} )}{200} : cos (\frac{π t}{12})

\frac{200 cos ( \frac{π t}{12} )}{200}

Divide the numbers:

\frac{200}{200} = 1

= cos (\frac{π t}{12})

Simplify

\frac{- 100}{200} : - \frac{1}{2}

\frac{- 100}{200}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{100}{200}

Cancel the common factor:

100

= - \frac{1}{2}

cos (\frac{π t}{12}) = - \frac{1}{2}

cos (\frac{π t}{12}) = - \frac{1}{2}

cos (\frac{π t}{12}) = - \frac{1}{2}

General solutions for

cos (\frac{π t}{12}) = - \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{π t}{12} = \frac{2 π}{3} + 2 πn, \frac{π t}{12} = \frac{4 π}{3} + 2 πn

\frac{π t}{12} = \frac{2 π}{3} + 2 πn, \frac{π t}{12} = \frac{4 π}{3} + 2 πn

Solve

\frac{π t}{12} = \frac{2 π}{3} + 2 πn : t = 8 + 24 n

\frac{π t}{12} = \frac{2 π}{3} + 2 πn

Multiply both sides by

12

\frac{π t}{12} = \frac{2 π}{3} + 2 πn

Multiply both sides by

12

\frac{12 π t}{12} = 12 \cdot \frac{2 π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π t}{12} = 12 \cdot \frac{2 π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π t}{12} : π t

\frac{12 π t}{12}

Divide the numbers:

\frac{12}{12} = 1

= π t

Simplify

12 \cdot \frac{2 π}{3} + 12 \cdot 2 πn : 8 π + 24 πn

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

12 \cdot \frac{2 π}{3} = 8 π

12 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 12}{3}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{24 π}{3}

Divide the numbers:

\frac{24}{3} = 8

= 8 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 8 π + 24 πn

π t = 8 π + 24 πn

π t = 8 π + 24 πn

π t = 8 π + 24 πn

Divide both sides by

π

π t = 8 π + 24 πn

Divide both sides by

π

\frac{π t}{π} = \frac{8 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} = \frac{8 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{8 π}{π} + \frac{24 πn}{π} : 8 + 24 n

\frac{8 π}{π} + \frac{24 πn}{π}

Cancel

\frac{8 π}{π} : 8

\frac{8 π}{π}

Cancel the common factor:

π

= 8

= 8 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 8 + 24 n

t = 8 + 24 n

t = 8 + 24 n

t = 8 + 24 n

Solve

\frac{π t}{12} = \frac{4 π}{3} + 2 πn : t = 16 + 24 n

\frac{π t}{12} = \frac{4 π}{3} + 2 πn

Multiply both sides by

12

\frac{π t}{12} = \frac{4 π}{3} + 2 πn

Multiply both sides by

12

\frac{12 π t}{12} = 12 \cdot \frac{4 π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π t}{12} = 12 \cdot \frac{4 π}{3} + 12 \cdot 2 πn

Simplify

\frac{12 π t}{12} : π t

\frac{12 π t}{12}

Divide the numbers:

\frac{12}{12} = 1

= π t

Simplify

12 \cdot \frac{4 π}{3} + 12 \cdot 2 πn : 16 π + 24 πn

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

12 \cdot \frac{4 π}{3} = 16 π

12 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 12}{3}

Multiply the numbers:

4 \cdot 12 = 48

= \frac{48 π}{3}

Divide the numbers:

\frac{48}{3} = 16

= 16 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 16 π + 24 πn

π t = 16 π + 24 πn

π t = 16 π + 24 πn

π t = 16 π + 24 πn

Divide both sides by

π

π t = 16 π + 24 πn

Divide both sides by

π

\frac{π t}{π} = \frac{16 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} = \frac{16 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{16 π}{π} + \frac{24 πn}{π} : 16 + 24 n

\frac{16 π}{π} + \frac{24 πn}{π}

Cancel

\frac{16 π}{π} : 16

\frac{16 π}{π}

Cancel the common factor:

π

= 16

= 16 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 16 + 24 n

t = 16 + 24 n

t = 16 + 24 n

t = 16 + 24 n

t = 8 + 24 n, t = 16 + 24 n

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Frequently Asked Questions (FAQ)

What is the general solution for 300=400+200cos((pit)/(12)) ?
The general solution for 300=400+200cos((pit)/(12)) is t=8+24n,t=16+24n