What is the general solution for 900=400sin((2pit}{365}+\frac{7pi)/8)+500 ?

The general solution for 900=400sin((2pit}{365}+\frac{7pi)/8)+500 is t=365n-1095/16

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Popular Trigonometry >

900=400sin((2pit}{365}+\frac{7pi)/8)+500

Solution

900 = 400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) + 500

Solution

t = 365 n - \frac{1095}{16}

Degrees

t = - 3921.17991 \dots^{\circ} + 20912.95952 \dots^{\circ} n

Solution steps

900 = 400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) + 500

Switch sides

400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) + 500 = 900

Move

500

to the right side

400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) + 500 = 900

Subtract

500

from both sides

400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) + 500 - 500 = 900 - 500

Simplify

400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) = 400

400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) = 400

Divide both sides by

400

400 sin (\frac{2 π t}{365} + \frac{7 π}{8}) = 400

Divide both sides by

400

\frac{400 sin ( \frac{2 π t}{365} + \frac{7 π}{8} )}{400} = \frac{400}{400}

Simplify

sin (\frac{2 π t}{365} + \frac{7 π}{8}) = 1

sin (\frac{2 π t}{365} + \frac{7 π}{8}) = 1

General solutions for

sin (\frac{2 π t}{365} + \frac{7 π}{8}) = 1

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{2 π t}{365} + \frac{7 π}{8} = \frac{π}{2} + 2 πn

\frac{2 π t}{365} + \frac{7 π}{8} = \frac{π}{2} + 2 πn

Solve

\frac{2 π t}{365} + \frac{7 π}{8} = \frac{π}{2} + 2 πn : t = 365 n - \frac{1095}{16}

\frac{2 π t}{365} + \frac{7 π}{8} = \frac{π}{2} + 2 πn

Move

\frac{7 π}{8}

to the right side

\frac{2 π t}{365} + \frac{7 π}{8} = \frac{π}{2} + 2 πn

Subtract

\frac{7 π}{8}

from both sides

\frac{2 π t}{365} + \frac{7 π}{8} - \frac{7 π}{8} = \frac{π}{2} + 2 πn - \frac{7 π}{8}

Simplify

\frac{2 π t}{365} + \frac{7 π}{8} - \frac{7 π}{8} = \frac{π}{2} + 2 πn - \frac{7 π}{8}

Simplify

\frac{2 π t}{365} + \frac{7 π}{8} - \frac{7 π}{8} : \frac{2 π t}{365}

\frac{2 π t}{365} + \frac{7 π}{8} - \frac{7 π}{8}

Add similar elements:

\frac{7 π}{8} - \frac{7 π}{8} = 0

= \frac{2 π t}{365}

Simplify

\frac{π}{2} + 2 πn - \frac{7 π}{8} : 2 πn - \frac{3 π}{8}

\frac{π}{2} + 2 πn - \frac{7 π}{8}

Group like terms

= 2 πn + \frac{π}{2} - \frac{7 π}{8}

Least Common Multiplier of

2, 8 : 8

2, 8

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

8

For

\frac{π}{2} :

multiply the denominator and numerator by

4

\frac{π}{2} = \frac{π 4}{2 \cdot 4} = \frac{π 4}{8}

= \frac{π 4}{8} - \frac{7 π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 - 7 π}{8}

Add similar elements:

4 π - 7 π = - 3 π

= \frac{- 3 π}{8}

Apply the fraction rule:

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

= 2 πn - \frac{3 π}{8}

\frac{2 π t}{365} = 2 πn - \frac{3 π}{8}

\frac{2 π t}{365} = 2 πn - \frac{3 π}{8}

\frac{2 π t}{365} = 2 πn - \frac{3 π}{8}

Multiply both sides by

365

\frac{2 π t}{365} = 2 πn - \frac{3 π}{8}

Multiply both sides by

365

\frac{365 \cdot 2 π t}{365} = 365 \cdot 2 πn - 365 \cdot \frac{3 π}{8}

Simplify

\frac{365 \cdot 2 π t}{365} = 365 \cdot 2 πn - 365 \cdot \frac{3 π}{8}

Simplify

\frac{365 \cdot 2 π t}{365} : 2 π t

\frac{365 \cdot 2 π t}{365}

Multiply the numbers:

365 \cdot 2 = 730

= \frac{730 π t}{365}

Divide the numbers:

\frac{730}{365} = 2

= 2 π t

Simplify

365 \cdot 2 πn - 365 \cdot \frac{3 π}{8} : 730 πn - \frac{1095 π}{8}

365 \cdot 2 πn - 365 \cdot \frac{3 π}{8}

365 \cdot 2 πn = 730 πn

365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= 730 πn

365 \cdot \frac{3 π}{8} = \frac{1095 π}{8}

365 \cdot \frac{3 π}{8}

Multiply fractions:

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

= \frac{3 π 365}{8}

Multiply the numbers:

3 \cdot 365 = 1095

= \frac{1095 π}{8}

= 730 πn - \frac{1095 π}{8}

2 π t = 730 πn - \frac{1095 π}{8}

2 π t = 730 πn - \frac{1095 π}{8}

2 π t = 730 πn - \frac{1095 π}{8}

Divide both sides by

2 π

2 π t = 730 πn - \frac{1095 π}{8}

Divide both sides by

2 π

\frac{2 π t}{2 π} = \frac{730 πn}{2 π} - \frac{\frac{1095 π}{8}}{2 π}

Simplify

\frac{2 π t}{2 π} = \frac{730 πn}{2 π} - \frac{\frac{1095 π}{8}}{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{730 πn}{2 π} - \frac{\frac{1095 π}{8}}{2 π} : 365 n - \frac{1095}{16}

\frac{730 πn}{2 π} - \frac{\frac{1095 π}{8}}{2 π}

\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{\frac{1095 π}{8}}{2 π} = \frac{1095}{16}

\frac{\frac{1095 π}{8}}{2 π}

Apply the fraction rule:

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

= \frac{1095 π}{8 \cdot 2 π}

Multiply the numbers:

8 \cdot 2 = 16

= \frac{1095 π}{16 π}

Cancel the common factor:

π

= \frac{1095}{16}

= 365 n - \frac{1095}{16}

t = 365 n - \frac{1095}{16}

t = 365 n - \frac{1095}{16}

t = 365 n - \frac{1095}{16}

t = 365 n - \frac{1095}{16}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 900=400sin((2pit}{365}+\frac{7pi)/8)+500 ?
The general solution for 900=400sin((2pit}{365}+\frac{7pi)/8)+500 is t=365n-1095/16