What is the general solution for sin((2pi(t+101.75))/(365))=-1 ?

The general solution for sin((2pi(t+101.75))/(365))=-1 is t=364.99999…n+172

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sin((2pi(t+101.75))/(365))=-1

Solution

sin (\frac{2 π ( t + 101.75 )}{365}) = - 1

Solution

t = 364.99999 \dots n + 172

Degrees

t = 9854.87407 \dots^{\circ} + 20912.95952 \dots^{\circ} n

Solution steps

sin (\frac{2 π ( t + 101.75 )}{365}) = - 1

General solutions for

sin (\frac{2 π ( t + 101.75 )}{365}) = - 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 + 101.75 )}{365} = \frac{3 π}{2} + 2 πn

\frac{2 π ( t + 101.75 )}{365} = \frac{3 π}{2} + 2 πn

Solve

\frac{2 π ( t + 101.75 )}{365} = \frac{3 π}{2} + 2 πn : t = 364.99999 \dots n + 172

\frac{2 π ( t + 101.75 )}{365} = \frac{3 π}{2} + 2 πn

Multiply both sides by

365

\frac{2 π ( t + 101.75 )}{365} = \frac{3 π}{2} + 2 πn

Multiply both sides by

365

\frac{365 \cdot 2 π ( t + 101.75 )}{365} = 365 \cdot \frac{3 π}{2} + 365 \cdot 2 πn

Simplify

\frac{365 \cdot 2 π ( t + 101.75 )}{365} = 365 \cdot \frac{3 π}{2} + 365 \cdot 2 πn

Simplify

\frac{365 \cdot 2 π ( t + 101.75 )}{365} : 6.28318 \dots (t + 101.75)

\frac{365 \cdot 2 π ( t + 101.75 )}{365}

Multiply the numbers:

365 \cdot 2 = 730

= \frac{730 π ( t + 101.75 )}{365}

Divide the numbers:

\frac{730}{365} = 2

= 2 π (t + 101.75)

Multiply the numbers:

2 \cdot 3.14159 \dots = 6.28318 \dots

= 6.28318 \dots (t + 101.75)

Simplify

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

365 \cdot \frac{3 π}{2} + 365 \cdot 2 πn

365 \cdot \frac{3 π}{2} = \frac{1095 π}{2}

365 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 365}{2}

Multiply the numbers:

3 \cdot 365 = 1095

= \frac{1095 π}{2}

365 \cdot 2 πn = 730 πn

365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= 730 πn

= \frac{1095 π}{2} + 730 πn

6.28318 \dots (t + 101.75) = \frac{1095 π}{2} + 730 πn

6.28318 \dots (t + 101.75) = \frac{1095 π}{2} + 730 πn

6.28318 \dots (t + 101.75) = \frac{1095 π}{2} + 730 πn

Divide both sides by

6.28318 \dots

6.28318 \dots (t + 101.75) = \frac{1095 π}{2} + 730 πn

Divide both sides by

6.28318 \dots

\frac{6.28318 \dots ( t + 101.75 )}{6.28318 \dots} = \frac{\frac{1095 π}{2}}{6.28318 \dots} + \frac{730 πn}{6.28318 \dots}

Simplify

\frac{6.28318 \dots ( t + 101.75 )}{6.28318 \dots} = \frac{\frac{1095 π}{2}}{6.28318 \dots} + \frac{730 πn}{6.28318 \dots}

Simplify

\frac{6.28318 \dots ( t + 101.75 )}{6.28318 \dots} : t + 101.75

\frac{6.28318 \dots ( t + 101.75 )}{6.28318 \dots}

Cancel the common factor:

6.28318 \dots

= t + 101.75

Simplify

\frac{\frac{1095 π}{2}}{6.28318 \dots} + \frac{730 πn}{6.28318 \dots} : 273.75 + 364.99999 \dots n

\frac{\frac{1095 π}{2}}{6.28318 \dots} + \frac{730 πn}{6.28318 \dots}

\frac{\frac{1095 π}{2}}{6.28318 \dots} = \frac{1095 π}{12.56637 \dots}

\frac{\frac{1095 π}{2}}{6.28318 \dots}

Apply the fraction rule:

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

= \frac{1095 π}{2 \cdot 6.28318 \dots}

Multiply the numbers:

2 \cdot 6.28318 \dots = 12.56637 \dots

= \frac{1095 π}{12.56637 \dots}

= \frac{1095 π}{12.56637 \dots} + \frac{730 πn}{6.28318 \dots}

\frac{1095 π}{12.56637 \dots} = 273.75

\frac{1095 π}{12.56637 \dots}

Multiply the numbers:

1095 \cdot 3.14159 \dots = 3440.04395 \dots

= \frac{3440.04395 \dots}{12.56637 \dots}

Divide the numbers:

\frac{3440.04395 \dots}{12.56637 \dots} = 273.75

= 273.75

\frac{730 πn}{6.28318 \dots} = 364.99999 \dots n

\frac{730 πn}{6.28318 \dots}

Multiply the numbers:

730 \cdot 3.14159 \dots = 2293.36263 \dots

= \frac{2293.36263 \dots n}{6.28318 \dots}

Divide the numbers:

\frac{2293.36263 \dots}{6.28318 \dots} = 364.99999 \dots

= 364.99999 \dots n

= 273.75 + 364.99999 \dots n

t + 101.75 = 273.75 + 364.99999 \dots n

t + 101.75 = 273.75 + 364.99999 \dots n

t + 101.75 = 273.75 + 364.99999 \dots n

Move

101.75

to the right side

t + 101.75 = 273.75 + 364.99999 \dots n

Subtract

101.75

from both sides

t + 101.75 - 101.75 = 273.75 + 364.99999 \dots n - 101.75

Simplify

t = 364.99999 \dots n + 172

t = 364.99999 \dots n + 172

t = 364.99999 \dots n + 172

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Frequently Asked Questions (FAQ)

What is the general solution for sin((2pi(t+101.75))/(365))=-1 ?
The general solution for sin((2pi(t+101.75))/(365))=-1 is t=364.99999…n+172