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Popular >

105=100+30sin((12pi)/5 t)

Solution

105 = 100 + 30 sin (\frac{12 π}{5} t)

Solution

t = \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}, t = \frac{5}{12} - \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}

Degrees

t = 1.27245 \dots^{\circ} + 47.74648 \dots^{\circ} n, t = 22.60078 \dots^{\circ} + 47.74648 \dots^{\circ} n

Solution steps

105 = 100 + 30 sin (\frac{12 π}{5} t)

Switch sides

100 + 30 sin (\frac{12 π}{5} t) = 105

Move

100

to the right side

100 + 30 sin (\frac{12 π}{5} t) = 105

Subtract

100

from both sides

100 + 30 sin (\frac{12 π}{5} t) - 100 = 105 - 100

Simplify

30 sin (\frac{12 π}{5} t) = 5

30 sin (\frac{12 π}{5} t) = 5

Divide both sides by

30

30 sin (\frac{12 π}{5} t) = 5

Divide both sides by

30

\frac{30 sin ( \frac{12 π}{5} t )}{30} = \frac{5}{30}

Simplify

sin (\frac{12 π}{5} t) = \frac{1}{6}

sin (\frac{12 π}{5} t) = \frac{1}{6}

Apply trig inverse properties

sin (\frac{12 π}{5} t) = \frac{1}{6}

General solutions for

sin (\frac{12 π}{5} t) = \frac{1}{6}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn, \frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn, \frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

Solve

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn : t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

5

\frac{12 π}{5} t = arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

5

5 \cdot \frac{12 π}{5} t = 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{12 π}{5} t = 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{12 π}{5} t : 12 π t

5 \cdot \frac{12 π}{5} t

Multiply fractions:

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

= \frac{12 \cdot 5 π}{5} t

Cancel the common factor:

5

= t \cdot 12 π

Simplify

5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn : 5 arcsin (\frac{1}{6}) + 10 πn

5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Multiply the numbers:

5 \cdot 2 = 10

= 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

Divide both sides by

12 π

12 π t = 5 arcsin (\frac{1}{6}) + 10 πn

Divide both sides by

12 π

\frac{12 π t}{12 π} = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplify

\frac{12 π t}{12 π} = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplify

\frac{12 π t}{12 π} : t

\frac{12 π t}{12 π}

Divide the numbers:

\frac{12}{12} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π} : \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Cancel

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Cancel

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Cancel the common factor:

2

= \frac{5 πn}{6 π}

Cancel the common factor:

π

= \frac{5 n}{6}

= \frac{5 n}{6}

= \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

Solve

\frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn : t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

5

\frac{12 π}{5} t = π - arcsin (\frac{1}{6}) + 2 πn

Multiply both sides by

5

5 \cdot \frac{12 π}{5} t = 5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{12 π}{5} t = 5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{12 π}{5} t : 12 π t

5 \cdot \frac{12 π}{5} t

Multiply fractions:

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

= \frac{12 \cdot 5 π}{5} t

Cancel the common factor:

5

= t \cdot 12 π

Simplify

5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn : 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

5 π - 5 arcsin (\frac{1}{6}) + 5 \cdot 2 πn

Multiply the numbers:

5 \cdot 2 = 10

= 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

Divide both sides by

12 π

12 π t = 5 π - 5 arcsin (\frac{1}{6}) + 10 πn

Divide both sides by

12 π

\frac{12 π t}{12 π} = \frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplify

\frac{12 π t}{12 π} = \frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Simplify

\frac{12 π t}{12 π} : t

\frac{12 π t}{12 π}

Divide the numbers:

\frac{12}{12} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π} : \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

\frac{5 π}{12 π} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Cancel

\frac{5 π}{12 π} : \frac{5}{12}

\frac{5 π}{12 π}

Cancel the common factor:

π

= \frac{5}{12}

= \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{10 πn}{12 π}

Cancel

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Cancel

\frac{10 πn}{12 π} : \frac{5 n}{6}

\frac{10 πn}{12 π}

Cancel the common factor:

2

= \frac{5 πn}{6 π}

Cancel the common factor:

π

= \frac{5 n}{6}

= \frac{5 n}{6}

= \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

t = \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}, t = \frac{5}{12} - \frac{5 arcsin ( \frac{1}{6} )}{12 π} + \frac{5 n}{6}

Show solutions in decimal form

t = \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}, t = \frac{5}{12} - \frac{5 \cdot 0.16744 \dots}{12 π} + \frac{5 n}{6}

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