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11=12+3.5sin((2pi)/(365)t)

Solution

11 = 12 + 3.5 sin (\frac{2 π}{365} t)

Solution

t = - \frac{365 \cdot 0.28975 \dots}{2 π} + 365 n, t = \frac{365}{2} + \frac{365 \cdot 0.28975 \dots}{2 π} + 365 n

Degrees

t = - 964.40981 \dots^{\circ} + 20912.95952 \dots^{\circ} n, t = 11420.88957 \dots^{\circ} + 20912.95952 \dots^{\circ} n

Solution steps

11 = 12 + 3.5 sin (\frac{2 π}{365} t)

Switch sides

12 + 3.5 sin (\frac{2 π}{365} t) = 11

Multiply both sides by

10

12 + 3.5 sin (\frac{2 π}{365} t) = 11

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

12 \cdot 10 + 3.5 sin (\frac{2 π}{365} t) \cdot 10 = 11 \cdot 10

Refine

120 + 35 sin (\frac{2 π}{365} t) = 110

120 + 35 sin (\frac{2 π}{365} t) = 110

Move

120

to the right side

120 + 35 sin (\frac{2 π}{365} t) = 110

Subtract

120

from both sides

120 + 35 sin (\frac{2 π}{365} t) - 120 = 110 - 120

Simplify

35 sin (\frac{2 π}{365} t) = - 10

35 sin (\frac{2 π}{365} t) = - 10

Divide both sides by

35

35 sin (\frac{2 π}{365} t) = - 10

Divide both sides by

35

\frac{35 sin ( \frac{2 π}{365} t )}{35} = \frac{- 10}{35}

Simplify

\frac{35 sin ( \frac{2 π}{365} t )}{35} = \frac{- 10}{35}

Simplify

\frac{35 sin ( \frac{2 π}{365} t )}{35} : sin (\frac{2 π}{365} t)

\frac{35 sin ( \frac{2 π}{365} t )}{35}

Divide the numbers:

\frac{35}{35} = 1

= sin (\frac{2 π}{365} t)

Simplify

\frac{- 10}{35} : - \frac{2}{7}

\frac{- 10}{35}

Apply the fraction rule:

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

= - \frac{10}{35}

Cancel the common factor:

5

= - \frac{2}{7}

sin (\frac{2 π}{365} t) = - \frac{2}{7}

sin (\frac{2 π}{365} t) = - \frac{2}{7}

sin (\frac{2 π}{365} t) = - \frac{2}{7}

Apply trig inverse properties

sin (\frac{2 π}{365} t) = - \frac{2}{7}

General solutions for

sin (\frac{2 π}{365} t) = - \frac{2}{7}

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

\frac{2 π}{365} t = arcsin (- \frac{2}{7}) + 2 πn, \frac{2 π}{365} t = π + arcsin (\frac{2}{7}) + 2 πn

\frac{2 π}{365} t = arcsin (- \frac{2}{7}) + 2 πn, \frac{2 π}{365} t = π + arcsin (\frac{2}{7}) + 2 πn

Solve

\frac{2 π}{365} t = arcsin (- \frac{2}{7}) + 2 πn : t = - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

\frac{2 π}{365} t = arcsin (- \frac{2}{7}) + 2 πn

Simplify

arcsin (- \frac{2}{7}) + 2 πn : - arcsin (\frac{2}{7}) + 2 πn

arcsin (- \frac{2}{7}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{7}) = - arcsin (\frac{2}{7})

= - arcsin (\frac{2}{7}) + 2 πn

\frac{2 π}{365} t = - arcsin (\frac{2}{7}) + 2 πn

Multiply both sides by

365

\frac{2 π}{365} t = - arcsin (\frac{2}{7}) + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} t = - 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t = - 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t : 2 π t

365 \cdot \frac{2 π}{365} t

Multiply fractions:

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

= \frac{2 \cdot 365 π}{365} t

Cancel the common factor:

365

= t \cdot 2 π

Simplify

- 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn : - 365 arcsin (\frac{2}{7}) + 730 πn

- 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= - 365 arcsin (\frac{2}{7}) + 730 πn

2 π t = - 365 arcsin (\frac{2}{7}) + 730 πn

2 π t = - 365 arcsin (\frac{2}{7}) + 730 πn

2 π t = - 365 arcsin (\frac{2}{7}) + 730 πn

Divide both sides by

2 π

2 π t = - 365 arcsin (\frac{2}{7}) + 730 πn

Divide both sides by

2 π

\frac{2 π t}{2 π} = - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} = - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} : t

\frac{2 π t}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

- \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π} : - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

- \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Divide the numbers:

\frac{730}{2} = 365

= \frac{365 πn}{π}

Cancel the common factor:

π

= 365 n

= 365 n

= - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

t = - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

t = - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

t = - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

Solve

\frac{2 π}{365} t = π + arcsin (\frac{2}{7}) + 2 πn : t = \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

\frac{2 π}{365} t = π + arcsin (\frac{2}{7}) + 2 πn

Multiply both sides by

365

\frac{2 π}{365} t = π + arcsin (\frac{2}{7}) + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} t = 365 π + 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t = 365 π + 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} t : 2 π t

365 \cdot \frac{2 π}{365} t

Multiply fractions:

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

= \frac{2 \cdot 365 π}{365} t

Cancel the common factor:

365

= t \cdot 2 π

Simplify

365 π + 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn : 365 π + 365 arcsin (\frac{2}{7}) + 730 πn

365 π + 365 arcsin (\frac{2}{7}) + 365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= 365 π + 365 arcsin (\frac{2}{7}) + 730 πn

2 π t = 365 π + 365 arcsin (\frac{2}{7}) + 730 πn

2 π t = 365 π + 365 arcsin (\frac{2}{7}) + 730 πn

2 π t = 365 π + 365 arcsin (\frac{2}{7}) + 730 πn

Divide both sides by

2 π

2 π t = 365 π + 365 arcsin (\frac{2}{7}) + 730 πn

Divide both sides by

2 π

\frac{2 π t}{2 π} = \frac{365 π}{2 π} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} = \frac{365 π}{2 π} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π t}{2 π} : t

\frac{2 π t}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{365 π}{2 π} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π} : \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

\frac{365 π}{2 π} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{365 π}{2 π} : \frac{365}{2}

\frac{365 π}{2 π}

Cancel the common factor:

π

= \frac{365}{2}

= \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Divide the numbers:

\frac{730}{2} = 365

= \frac{365 πn}{π}

Cancel the common factor:

π

= 365 n

= 365 n

= \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

t = \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

t = \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

t = \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

t = - \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n, t = \frac{365}{2} + \frac{365 arcsin ( \frac{2}{7} )}{2 π} + 365 n

Show solutions in decimal form

t = - \frac{365 \cdot 0.28975 \dots}{2 π} + 365 n, t = \frac{365}{2} + \frac{365 \cdot 0.28975 \dots}{2 π} + 365 n

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