What is the general solution for 25=-20cos((2pi)/(365)(x+10))+25 ?

The general solution for 25=-20cos((2pi)/(365)(x+10))+25 is x=365n+325/4 ,x=365n+1055/4

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25=-20cos((2pi)/(365)(x+10))+25

Solution

25 = - 20 cos (\frac{2 π}{365} (x + 10)) + 25

Solution

x = 365 n + \frac{325}{4}, x = 365 n + \frac{1055}{4}

Degrees

x = 4655.28208 \dots^{\circ} + 20912.95952 \dots^{\circ} n, x = 15111.76184 \dots^{\circ} + 20912.95952 \dots^{\circ} n

Solution steps

25 = - 20 cos (\frac{2 π}{365} (x + 10)) + 25

Switch sides

- 20 cos (\frac{2 π}{365} (x + 10)) + 25 = 25

Subtract

25

from both sides

- 20 cos (\frac{2 π}{365} (x + 10)) + 25 - 25 = 25 - 25

Simplify

- 20 cos (\frac{2 π}{365} (x + 10)) = 0

Divide both sides by

- 20

- 20 cos (\frac{2 π}{365} (x + 10)) = 0

Divide both sides by

- 20

\frac{- 20 cos ( \frac{2 π}{365} ( x + 10 ) )}{- 20} = \frac{0}{- 20}

Simplify

cos (\frac{2 π}{365} (x + 10)) = 0

cos (\frac{2 π}{365} (x + 10)) = 0

General solutions for

cos (\frac{2 π}{365} (x + 10)) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{2 π}{365} (x + 10) = \frac{π}{2} + 2 πn, \frac{2 π}{365} (x + 10) = \frac{3 π}{2} + 2 πn

\frac{2 π}{365} (x + 10) = \frac{π}{2} + 2 πn, \frac{2 π}{365} (x + 10) = \frac{3 π}{2} + 2 πn

Solve

\frac{2 π}{365} (x + 10) = \frac{π}{2} + 2 πn : x = 365 n + \frac{325}{4}

\frac{2 π}{365} (x + 10) = \frac{π}{2} + 2 πn

Multiply both sides by

365

\frac{2 π}{365} (x + 10) = \frac{π}{2} + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} (x + 10) = 365 \cdot \frac{π}{2} + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x + 10) = 365 \cdot \frac{π}{2} + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x + 10) : 2 π (x + 10)

365 \cdot \frac{2 π}{365} (x + 10)

Multiply fractions:

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

= \frac{2 \cdot 365 π}{365} (x + 10)

Cancel the common factor:

365

= (x + 10) \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 (x + 10) = \frac{365 π}{2} + 730 πn

2 π (x + 10) = \frac{365 π}{2} + 730 πn

2 π (x + 10) = \frac{365 π}{2} + 730 πn

Divide both sides by

2 π

2 π (x + 10) = \frac{365 π}{2} + 730 πn

Divide both sides by

2 π

\frac{2 π ( x + 10 )}{2 π} = \frac{\frac{365 π}{2}}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x + 10 )}{2 π} = \frac{\frac{365 π}{2}}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x + 10 )}{2 π} : x + 10

\frac{2 π ( x + 10 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( x + 10 )}{π}

Cancel the common factor:

π

= x + 10

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

x + 10 = \frac{365}{4} + 365 n

x + 10 = \frac{365}{4} + 365 n

x + 10 = \frac{365}{4} + 365 n

Move

10

to the right side

x + 10 = \frac{365}{4} + 365 n

Subtract

10

from both sides

x + 10 - 10 = \frac{365}{4} + 365 n - 10

Simplify

x + 10 - 10 = \frac{365}{4} + 365 n - 10

Simplify

x + 10 - 10 : x

x + 10 - 10

Add similar elements:

10 - 10 = 0

= x

Simplify

\frac{365}{4} + 365 n - 10 : 365 n + \frac{325}{4}

\frac{365}{4} + 365 n - 10

Combine the fractions

- 10 + \frac{365}{4} : \frac{325}{4}

- 10 + \frac{365}{4}

Convert element to fraction:

10 = \frac{10 \cdot 4}{4}

= - \frac{10 \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{- 10 \cdot 4 + 365}{4}

- 10 \cdot 4 + 365 = 325

- 10 \cdot 4 + 365

Multiply the numbers:

10 \cdot 4 = 40

= - 40 + 365

Add/Subtract the numbers:

- 40 + 365 = 325

= 325

= \frac{325}{4}

= 365 n + \frac{325}{4}

x = 365 n + \frac{325}{4}

x = 365 n + \frac{325}{4}

x = 365 n + \frac{325}{4}

Solve

\frac{2 π}{365} (x + 10) = \frac{3 π}{2} + 2 πn : x = 365 n + \frac{1055}{4}

\frac{2 π}{365} (x + 10) = \frac{3 π}{2} + 2 πn

Multiply both sides by

365

\frac{2 π}{365} (x + 10) = \frac{3 π}{2} + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} (x + 10) = 365 \cdot \frac{3 π}{2} + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x + 10) = 365 \cdot \frac{3 π}{2} + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x + 10) : 2 π (x + 10)

365 \cdot \frac{2 π}{365} (x + 10)

Multiply fractions:

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

= \frac{2 \cdot 365 π}{365} (x + 10)

Cancel the common factor:

365

= (x + 10) \cdot 2 π

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

2 π (x + 10) = \frac{1095 π}{2} + 730 πn

2 π (x + 10) = \frac{1095 π}{2} + 730 πn

2 π (x + 10) = \frac{1095 π}{2} + 730 πn

Divide both sides by

2 π

2 π (x + 10) = \frac{1095 π}{2} + 730 πn

Divide both sides by

2 π

\frac{2 π ( x + 10 )}{2 π} = \frac{\frac{1095 π}{2}}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x + 10 )}{2 π} = \frac{\frac{1095 π}{2}}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x + 10 )}{2 π} : x + 10

\frac{2 π ( x + 10 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( x + 10 )}{π}

Cancel the common factor:

π

= x + 10

Simplify

\frac{\frac{1095 π}{2}}{2 π} + \frac{730 πn}{2 π} : \frac{1095}{4} + 365 n

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

\frac{\frac{1095 π}{2}}{2 π} = \frac{1095}{4}

\frac{\frac{1095 π}{2}}{2 π}

Apply the fraction rule:

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

= \frac{1095 π}{2 \cdot 2 π}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1095 π}{4 π}

Cancel the common factor:

π

= \frac{1095}{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{1095}{4} + 365 n

x + 10 = \frac{1095}{4} + 365 n

x + 10 = \frac{1095}{4} + 365 n

x + 10 = \frac{1095}{4} + 365 n

Move

10

to the right side

x + 10 = \frac{1095}{4} + 365 n

Subtract

10

from both sides

x + 10 - 10 = \frac{1095}{4} + 365 n - 10

Simplify

x + 10 - 10 = \frac{1095}{4} + 365 n - 10

Simplify

x + 10 - 10 : x

x + 10 - 10

Add similar elements:

10 - 10 = 0

= x

Simplify

\frac{1095}{4} + 365 n - 10 : 365 n + \frac{1055}{4}

\frac{1095}{4} + 365 n - 10

Combine the fractions

- 10 + \frac{1095}{4} : \frac{1055}{4}

- 10 + \frac{1095}{4}

Convert element to fraction:

10 = \frac{10 \cdot 4}{4}

= - \frac{10 \cdot 4}{4} + \frac{1095}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- 10 \cdot 4 + 1095}{4}

- 10 \cdot 4 + 1095 = 1055

- 10 \cdot 4 + 1095

Multiply the numbers:

10 \cdot 4 = 40

= - 40 + 1095

Add/Subtract the numbers:

- 40 + 1095 = 1055

= 1055

= \frac{1055}{4}

= 365 n + \frac{1055}{4}

x = 365 n + \frac{1055}{4}

x = 365 n + \frac{1055}{4}

x = 365 n + \frac{1055}{4}

x = 365 n + \frac{325}{4}, x = 365 n + \frac{1055}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for 25=-20cos((2pi)/(365)(x+10))+25 ?
The general solution for 25=-20cos((2pi)/(365)(x+10))+25 is x=365n+325/4 ,x=365n+1055/4