What is the general solution for solvefor t,10=14+8sin((pit)/(12)) ?

The general solution for solvefor t,10=14+8sin((pit)/(12)) is t=14+24n,t=22+24n

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solvefor t,10=14+8sin((pit)/(12))

Solution

solve for

t, 10 = 14 + 8 sin (\frac{π t}{12})

Solution

t = 14 + 24 n, t = 22 + 24 n

Degrees

t = 802.14091 \dots^{\circ} + 1375.09870 \dots^{\circ} n, t = 1260.50714 \dots^{\circ} + 1375.09870 \dots^{\circ} n

Solution steps

10 = 14 + 8 sin (\frac{π t}{12})

Switch sides

14 + 8 sin (\frac{π t}{12}) = 10

Move

14

to the right side

14 + 8 sin (\frac{π t}{12}) = 10

Subtract

14

from both sides

14 + 8 sin (\frac{π t}{12}) - 14 = 10 - 14

Simplify

8 sin (\frac{π t}{12}) = - 4

8 sin (\frac{π t}{12}) = - 4

Divide both sides by

8

8 sin (\frac{π t}{12}) = - 4

Divide both sides by

8

\frac{8 sin ( \frac{π t}{12} )}{8} = \frac{- 4}{8}

Simplify

\frac{8 sin ( \frac{π t}{12} )}{8} = \frac{- 4}{8}

Simplify

\frac{8 sin ( \frac{π t}{12} )}{8} : sin (\frac{π t}{12})

\frac{8 sin ( \frac{π t}{12} )}{8}

Divide the numbers:

\frac{8}{8} = 1

= sin (\frac{π t}{12})

Simplify

\frac{- 4}{8} : - \frac{1}{2}

\frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

sin (\frac{π t}{12}) = - \frac{1}{2}

sin (\frac{π t}{12}) = - \frac{1}{2}

sin (\frac{π t}{12}) = - \frac{1}{2}

General solutions for

sin (\frac{π t}{12}) = - \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{π t}{12} = \frac{7 π}{6} + 2 πn, \frac{π t}{12} = \frac{11 π}{6} + 2 πn

\frac{π t}{12} = \frac{7 π}{6} + 2 πn, \frac{π t}{12} = \frac{11 π}{6} + 2 πn

Solve

\frac{π t}{12} = \frac{7 π}{6} + 2 πn : t = 14 + 24 n

\frac{π t}{12} = \frac{7 π}{6} + 2 πn

Multiply both sides by

12

\frac{π t}{12} = \frac{7 π}{6} + 2 πn

Multiply both sides by

12

\frac{12 π t}{12} = 12 \cdot \frac{7 π}{6} + 12 \cdot 2 πn

Simplify

\frac{12 π t}{12} = 12 \cdot \frac{7 π}{6} + 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{7 π}{6} + 12 \cdot 2 πn : 14 π + 24 πn

12 \cdot \frac{7 π}{6} + 12 \cdot 2 πn

12 \cdot \frac{7 π}{6} = 14 π

12 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 12}{6}

Multiply the numbers:

7 \cdot 12 = 84

= \frac{84 π}{6}

Divide the numbers:

\frac{84}{6} = 14

= 14 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 14 π + 24 πn

π t = 14 π + 24 πn

π t = 14 π + 24 πn

π t = 14 π + 24 πn

Divide both sides by

π

π t = 14 π + 24 πn

Divide both sides by

π

\frac{π t}{π} = \frac{14 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} = \frac{14 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{14 π}{π} + \frac{24 πn}{π} : 14 + 24 n

\frac{14 π}{π} + \frac{24 πn}{π}

Cancel

\frac{14 π}{π} : 14

\frac{14 π}{π}

Cancel the common factor:

π

= 14

= 14 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 14 + 24 n

t = 14 + 24 n

t = 14 + 24 n

t = 14 + 24 n

Solve

\frac{π t}{12} = \frac{11 π}{6} + 2 πn : t = 22 + 24 n

\frac{π t}{12} = \frac{11 π}{6} + 2 πn

Multiply both sides by

12

\frac{π t}{12} = \frac{11 π}{6} + 2 πn

Multiply both sides by

12

\frac{12 π t}{12} = 12 \cdot \frac{11 π}{6} + 12 \cdot 2 πn

Simplify

\frac{12 π t}{12} = 12 \cdot \frac{11 π}{6} + 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{11 π}{6} + 12 \cdot 2 πn : 22 π + 24 πn

12 \cdot \frac{11 π}{6} + 12 \cdot 2 πn

12 \cdot \frac{11 π}{6} = 22 π

12 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 12}{6}

Multiply the numbers:

11 \cdot 12 = 132

= \frac{132 π}{6}

Divide the numbers:

\frac{132}{6} = 22

= 22 π

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

= 22 π + 24 πn

π t = 22 π + 24 πn

π t = 22 π + 24 πn

π t = 22 π + 24 πn

Divide both sides by

π

π t = 22 π + 24 πn

Divide both sides by

π

\frac{π t}{π} = \frac{22 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} = \frac{22 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{22 π}{π} + \frac{24 πn}{π} : 22 + 24 n

\frac{22 π}{π} + \frac{24 πn}{π}

Cancel

\frac{22 π}{π} : 22

\frac{22 π}{π}

Cancel the common factor:

π

= 22

= 22 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 22 + 24 n

t = 22 + 24 n

t = 22 + 24 n

t = 22 + 24 n

t = 14 + 24 n, t = 22 + 24 n

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor t,10=14+8sin((pit)/(12)) ?
The general solution for solvefor t,10=14+8sin((pit)/(12)) is t=14+24n,t=22+24n