What is the general solution for 50=36sin((2pi)/(365)(t-101))+14 ?

The general solution for 50=36sin((2pi)/(365)(t-101))+14 is t=365n+769/4

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50=36sin((2pi)/(365)(t-101))+14

Solution

50 = 36 sin (\frac{2 π}{365} (t - 101)) + 14

Solution

t = 365 n + \frac{769}{4}

Degrees

t = 11015.11361 \dots^{\circ} + 20912.95952 \dots^{\circ} n

Solution steps

50 = 36 sin (\frac{2 π}{365} (t - 101)) + 14

Switch sides

36 sin (\frac{2 π}{365} (t - 101)) + 14 = 50

Move

14

to the right side

36 sin (\frac{2 π}{365} (t - 101)) + 14 = 50

Subtract

14

from both sides

36 sin (\frac{2 π}{365} (t - 101)) + 14 - 14 = 50 - 14

Simplify

36 sin (\frac{2 π}{365} (t - 101)) = 36

36 sin (\frac{2 π}{365} (t - 101)) = 36

Divide both sides by

36

36 sin (\frac{2 π}{365} (t - 101)) = 36

Divide both sides by

36

\frac{36 sin ( \frac{2 π}{365} ( t - 101 ) )}{36} = \frac{36}{36}

Simplify

sin (\frac{2 π}{365} (t - 101)) = 1

sin (\frac{2 π}{365} (t - 101)) = 1

General solutions for

sin (\frac{2 π}{365} (t - 101)) = 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}{365} (t - 101) = \frac{π}{2} + 2 πn

\frac{2 π}{365} (t - 101) = \frac{π}{2} + 2 πn

Solve

\frac{2 π}{365} (t - 101) = \frac{π}{2} + 2 πn : t = 365 n + \frac{769}{4}

\frac{2 π}{365} (t - 101) = \frac{π}{2} + 2 πn

Multiply both sides by

365

\frac{2 π}{365} (t - 101) = \frac{π}{2} + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} (t - 101) = 365 \cdot \frac{π}{2} + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (t - 101) = 365 \cdot \frac{π}{2} + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (t - 101) : 2 π (t - 101)

365 \cdot \frac{2 π}{365} (t - 101)

Multiply fractions:

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

= \frac{2 \cdot 365 π}{365} (t - 101)

Cancel the common factor:

365

= (t - 101) \cdot 2 π

Simplify

365 \cdot \frac{π}{2} + 365 \cdot 2 πn : \frac{365 π}{2} + 730 πn

365 \cdot \frac{π}{2} + 365 \cdot 2 πn

Multiply

365 \cdot \frac{π}{2} : \frac{365 π}{2}

365 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 365}{2}

= \frac{365 π}{2} + 365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= \frac{365 π}{2} + 730 πn

2 π (t - 101) = \frac{365 π}{2} + 730 πn

2 π (t - 101) = \frac{365 π}{2} + 730 πn

2 π (t - 101) = \frac{365 π}{2} + 730 πn

Divide both sides by

2 π

2 π (t - 101) = \frac{365 π}{2} + 730 πn

Divide both sides by

2 π

\frac{2 π ( t - 101 )}{2 π} = \frac{\frac{365 π}{2}}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( t - 101 )}{2 π} = \frac{\frac{365 π}{2}}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( t - 101 )}{2 π} : t - 101

\frac{2 π ( t - 101 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( t - 101 )}{π}

Cancel the common factor:

π

= t - 101

Simplify

\frac{\frac{365 π}{2}}{2 π} + \frac{730 πn}{2 π} : \frac{365}{4} + 365 n

\frac{\frac{365 π}{2}}{2 π} + \frac{730 πn}{2 π}

\frac{\frac{365 π}{2}}{2 π} = \frac{365}{4}

\frac{\frac{365 π}{2}}{2 π}

Apply the fraction rule:

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

= \frac{365 π}{2 \cdot 2 π}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{365 π}{4 π}

Cancel the common factor:

π

= \frac{365}{4}

\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}{4} + 365 n

t - 101 = \frac{365}{4} + 365 n

t - 101 = \frac{365}{4} + 365 n

t - 101 = \frac{365}{4} + 365 n

Move

101

to the right side

t - 101 = \frac{365}{4} + 365 n

Add

101

to both sides

t - 101 + 101 = \frac{365}{4} + 365 n + 101

Simplify

t - 101 + 101 = \frac{365}{4} + 365 n + 101

Simplify

t - 101 + 101 : t

t - 101 + 101

Add similar elements:

- 101 + 101 = 0

= t

Simplify

\frac{365}{4} + 365 n + 101 : 365 n + \frac{769}{4}

\frac{365}{4} + 365 n + 101

Combine the fractions

101 + \frac{365}{4} : \frac{769}{4}

101 + \frac{365}{4}

Convert element to fraction:

101 = \frac{101 \cdot 4}{4}

= \frac{101 \cdot 4}{4} + \frac{365}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{101 \cdot 4 + 365}{4}

101 \cdot 4 + 365 = 769

101 \cdot 4 + 365

Multiply the numbers:

101 \cdot 4 = 404

= 404 + 365

Add the numbers:

404 + 365 = 769

= 769

= \frac{769}{4}

= 365 n + \frac{769}{4}

t = 365 n + \frac{769}{4}

t = 365 n + \frac{769}{4}

t = 365 n + \frac{769}{4}

t = 365 n + \frac{769}{4}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 50=36sin((2pi)/(365)(t-101))+14 ?
The general solution for 50=36sin((2pi)/(365)(t-101))+14 is t=365n+769/4