What is the general solution for 0=(2pi^2cos(\frac{pix)/(10))}{25} ?

The general solution for 0=(2pi^2cos(\frac{pix)/(10))}{25} is x=5+20n,x=15+20n

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0=(2pi^2cos(\frac{pix)/(10))}{25}

Solution

0 = \frac{2 π ^{2} cos ( \frac{π x}{10} )}{25}

Solution

x = 5 + 20 n, x = 15 + 20 n

Degrees

x = 286.47889 \dots^{\circ} + 1145.91559 \dots^{\circ} n, x = 859.43669 \dots^{\circ} + 1145.91559 \dots^{\circ} n

Solution steps

0 = \frac{2 π ^{2} cos ( \frac{π x}{10} )}{25}

Switch sides

\frac{2 π ^{2} cos ( \frac{π x}{10} )}{25} = 0

Multiply both sides by

25

\frac{2 π ^{2} cos ( \frac{π x}{10} )}{25} = 0

Multiply both sides by

25

\frac{25 \cdot 2 π ^{2} cos ( \frac{π x}{10} )}{25} = 0 \cdot 25

Simplify

2 π^{2} cos (\frac{π x}{10}) = 0

2 π^{2} cos (\frac{π x}{10}) = 0

Divide both sides by

2 π^{2}

2 π^{2} cos (\frac{π x}{10}) = 0

Divide both sides by

2 π^{2}

\frac{2 π ^{2} cos ( \frac{π x}{10} )}{2 π ^{2}} = \frac{0}{2 π ^{2}}

Simplify

cos (\frac{π x}{10}) = 0

cos (\frac{π x}{10}) = 0

General solutions for

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

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

Solve

\frac{π x}{10} = \frac{π}{2} + 2 πn : x = 5 + 20 n

\frac{π x}{10} = \frac{π}{2} + 2 πn

Multiply both sides by

10

\frac{π x}{10} = \frac{π}{2} + 2 πn

Multiply both sides by

10

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

Simplify

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

Simplify

\frac{10 π x}{10} : π x

\frac{10 π x}{10}

Divide the numbers:

\frac{10}{10} = 1

= π x

Simplify

10 \cdot \frac{π}{2} + 10 \cdot 2 πn : 5 π + 20 πn

10 \cdot \frac{π}{2} + 10 \cdot 2 πn

10 \cdot \frac{π}{2} = 5 π

10 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 10}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 π

10 \cdot 2 πn = 20 πn

10 \cdot 2 πn

Multiply the numbers:

10 \cdot 2 = 20

= 20 πn

= 5 π + 20 πn

π x = 5 π + 20 πn

π x = 5 π + 20 πn

π x = 5 π + 20 πn

Divide both sides by

π

π x = 5 π + 20 πn

Divide both sides by

π

\frac{π x}{π} = \frac{5 π}{π} + \frac{20 πn}{π}

Simplify

\frac{π x}{π} = \frac{5 π}{π} + \frac{20 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{5 π}{π} + \frac{20 πn}{π} : 5 + 20 n

\frac{5 π}{π} + \frac{20 πn}{π}

Cancel

\frac{5 π}{π} : 5

\frac{5 π}{π}

Cancel the common factor:

π

= 5

= 5 + \frac{20 πn}{π}

Cancel

\frac{20 πn}{π} : 20 n

\frac{20 πn}{π}

Cancel the common factor:

π

= 20 n

= 5 + 20 n

x = 5 + 20 n

x = 5 + 20 n

x = 5 + 20 n

Solve

\frac{π x}{10} = \frac{3 π}{2} + 2 πn : x = 15 + 20 n

\frac{π x}{10} = \frac{3 π}{2} + 2 πn

Multiply both sides by

10

\frac{π x}{10} = \frac{3 π}{2} + 2 πn

Multiply both sides by

10

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

Simplify

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

Simplify

\frac{10 π x}{10} : π x

\frac{10 π x}{10}

Divide the numbers:

\frac{10}{10} = 1

= π x

Simplify

10 \cdot \frac{3 π}{2} + 10 \cdot 2 πn : 15 π + 20 πn

10 \cdot \frac{3 π}{2} + 10 \cdot 2 πn

10 \cdot \frac{3 π}{2} = 15 π

10 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 10}{2}

Multiply the numbers:

3 \cdot 10 = 30

= \frac{30 π}{2}

Divide the numbers:

\frac{30}{2} = 15

= 15 π

10 \cdot 2 πn = 20 πn

10 \cdot 2 πn

Multiply the numbers:

10 \cdot 2 = 20

= 20 πn

= 15 π + 20 πn

π x = 15 π + 20 πn

π x = 15 π + 20 πn

π x = 15 π + 20 πn

Divide both sides by

π

π x = 15 π + 20 πn

Divide both sides by

π

\frac{π x}{π} = \frac{15 π}{π} + \frac{20 πn}{π}

Simplify

\frac{π x}{π} = \frac{15 π}{π} + \frac{20 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{15 π}{π} + \frac{20 πn}{π} : 15 + 20 n

\frac{15 π}{π} + \frac{20 πn}{π}

Cancel

\frac{15 π}{π} : 15

\frac{15 π}{π}

Cancel the common factor:

π

= 15

= 15 + \frac{20 πn}{π}

Cancel

\frac{20 πn}{π} : 20 n

\frac{20 πn}{π}

Cancel the common factor:

π

= 20 n

= 15 + 20 n

x = 15 + 20 n

x = 15 + 20 n

x = 15 + 20 n

x = 5 + 20 n, x = 15 + 20 n

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Frequently Asked Questions (FAQ)

What is the general solution for 0=(2pi^2cos(\frac{pix)/(10))}{25} ?
The general solution for 0=(2pi^2cos(\frac{pix)/(10))}{25} is x=5+20n,x=15+20n