What is the general solution for 0=-(9pi^2)/(3200)cos((pix)/(80)) ?

The general solution for 0=-(9pi^2)/(3200)cos((pix)/(80)) is x=40+160n,x=120+160n

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0=-(9pi^2)/(3200)cos((pix)/(80))

Solution

0 = - \frac{9 π ^{2}}{3200} cos (\frac{π x}{80})

Solution

x = 40 + 160 n, x = 120 + 160 n

Degrees

x = 2291.83118 \dots^{\circ} + 9167.32472 \dots^{\circ} n, x = 6875.49354 \dots^{\circ} + 9167.32472 \dots^{\circ} n

Solution steps

0 = - \frac{9 π ^{2}}{3200} cos (\frac{π x}{80})

Switch sides

- \frac{9 π ^{2}}{3200} cos (\frac{π x}{80}) = 0

Multiply both sides by

3200

- \frac{9 π ^{2}}{3200} cos (\frac{π x}{80}) = 0

Multiply both sides by

3200

3200 (- \frac{9 π ^{2}}{3200} cos (\frac{π x}{80})) = 0 \cdot 3200

Simplify

- 9 π^{2} cos (\frac{π x}{80}) = 0

- 9 π^{2} cos (\frac{π x}{80}) = 0

Divide both sides by

- 9 π^{2}

- 9 π^{2} cos (\frac{π x}{80}) = 0

Divide both sides by

- 9 π^{2}

\frac{- 9 π ^{2} cos ( \frac{π x}{80} )}{- 9 π ^{2}} = \frac{0}{- 9 π ^{2}}

Simplify

cos (\frac{π x}{80}) = 0

cos (\frac{π x}{80}) = 0

General solutions for

cos (\frac{π x}{80}) = 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}{80} = \frac{π}{2} + 2 πn, \frac{π x}{80} = \frac{3 π}{2} + 2 πn

\frac{π x}{80} = \frac{π}{2} + 2 πn, \frac{π x}{80} = \frac{3 π}{2} + 2 πn

Solve

\frac{π x}{80} = \frac{π}{2} + 2 πn : x = 40 + 160 n

\frac{π x}{80} = \frac{π}{2} + 2 πn

Multiply both sides by

80

\frac{π x}{80} = \frac{π}{2} + 2 πn

Multiply both sides by

80

\frac{80 π x}{80} = 80 \cdot \frac{π}{2} + 80 \cdot 2 πn

Simplify

\frac{80 π x}{80} = 80 \cdot \frac{π}{2} + 80 \cdot 2 πn

Simplify

\frac{80 π x}{80} : π x

\frac{80 π x}{80}

Divide the numbers:

\frac{80}{80} = 1

= π x

Simplify

80 \cdot \frac{π}{2} + 80 \cdot 2 πn : 40 π + 160 πn

80 \cdot \frac{π}{2} + 80 \cdot 2 πn

80 \cdot \frac{π}{2} = 40 π

80 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 80}{2}

Divide the numbers:

\frac{80}{2} = 40

= 40 π

80 \cdot 2 πn = 160 πn

80 \cdot 2 πn

Multiply the numbers:

80 \cdot 2 = 160

= 160 πn

= 40 π + 160 πn

π x = 40 π + 160 πn

π x = 40 π + 160 πn

π x = 40 π + 160 πn

Divide both sides by

π

π x = 40 π + 160 πn

Divide both sides by

π

\frac{π x}{π} = \frac{40 π}{π} + \frac{160 πn}{π}

Simplify

\frac{π x}{π} = \frac{40 π}{π} + \frac{160 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{40 π}{π} + \frac{160 πn}{π} : 40 + 160 n

\frac{40 π}{π} + \frac{160 πn}{π}

Cancel

\frac{40 π}{π} : 40

\frac{40 π}{π}

Cancel the common factor:

π

= 40

= 40 + \frac{160 πn}{π}

Cancel

\frac{160 πn}{π} : 160 n

\frac{160 πn}{π}

Cancel the common factor:

π

= 160 n

= 40 + 160 n

x = 40 + 160 n

x = 40 + 160 n

x = 40 + 160 n

Solve

\frac{π x}{80} = \frac{3 π}{2} + 2 πn : x = 120 + 160 n

\frac{π x}{80} = \frac{3 π}{2} + 2 πn

Multiply both sides by

80

\frac{π x}{80} = \frac{3 π}{2} + 2 πn

Multiply both sides by

80

\frac{80 π x}{80} = 80 \cdot \frac{3 π}{2} + 80 \cdot 2 πn

Simplify

\frac{80 π x}{80} = 80 \cdot \frac{3 π}{2} + 80 \cdot 2 πn

Simplify

\frac{80 π x}{80} : π x

\frac{80 π x}{80}

Divide the numbers:

\frac{80}{80} = 1

= π x

Simplify

80 \cdot \frac{3 π}{2} + 80 \cdot 2 πn : 120 π + 160 πn

80 \cdot \frac{3 π}{2} + 80 \cdot 2 πn

80 \cdot \frac{3 π}{2} = 120 π

80 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 80}{2}

Multiply the numbers:

3 \cdot 80 = 240

= \frac{240 π}{2}

Divide the numbers:

\frac{240}{2} = 120

= 120 π

80 \cdot 2 πn = 160 πn

80 \cdot 2 πn

Multiply the numbers:

80 \cdot 2 = 160

= 160 πn

= 120 π + 160 πn

π x = 120 π + 160 πn

π x = 120 π + 160 πn

π x = 120 π + 160 πn

Divide both sides by

π

π x = 120 π + 160 πn

Divide both sides by

π

\frac{π x}{π} = \frac{120 π}{π} + \frac{160 πn}{π}

Simplify

\frac{π x}{π} = \frac{120 π}{π} + \frac{160 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{120 π}{π} + \frac{160 πn}{π} : 120 + 160 n

\frac{120 π}{π} + \frac{160 πn}{π}

Cancel

\frac{120 π}{π} : 120

\frac{120 π}{π}

Cancel the common factor:

π

= 120

= 120 + \frac{160 πn}{π}

Cancel

\frac{160 πn}{π} : 160 n

\frac{160 πn}{π}

Cancel the common factor:

π

= 160 n

= 120 + 160 n

x = 120 + 160 n

x = 120 + 160 n

x = 120 + 160 n

x = 40 + 160 n, x = 120 + 160 n

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Frequently Asked Questions (FAQ)

What is the general solution for 0=-(9pi^2)/(3200)cos((pix)/(80)) ?
The general solution for 0=-(9pi^2)/(3200)cos((pix)/(80)) is x=40+160n,x=120+160n