What is the general solution for pi/3 cos((pix)/6)=0,x ?

The general solution for pi/3 cos((pix)/6)=0,x is No Solution for x\in\mathbb{R}

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pi/3 cos((pix)/6)=0,x

Solution

\frac{π}{3} cos (\frac{π x}{6}) = 0, x

Solution

No Solution for x \in R

Solution steps

\frac{π}{3} cos (\frac{π x}{6}) = 0, x

Multiply both sides by

3

\frac{π}{3} cos (\frac{π x}{6}) = 0

Multiply both sides by

3

3 \cdot \frac{π}{3} cos (\frac{π x}{6}) = 0 \cdot 3

Simplify

π cos (\frac{π x}{6}) = 0

π cos (\frac{π x}{6}) = 0

Divide both sides by

π

π cos (\frac{π x}{6}) = 0

Divide both sides by

π

\frac{π cos ( \frac{π x}{6} )}{π} = \frac{0}{π}

Simplify

cos (\frac{π x}{6}) = 0

cos (\frac{π x}{6}) = 0

General solutions for

cos (\frac{π x}{6}) = 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}{6} = \frac{π}{2} + 2 πn, \frac{π x}{6} = \frac{3 π}{2} + 2 πn

\frac{π x}{6} = \frac{π}{2} + 2 πn, \frac{π x}{6} = \frac{3 π}{2} + 2 πn

Solve

\frac{π x}{6} = \frac{π}{2} + 2 πn : x = 3 + 12 n

\frac{π x}{6} = \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{π x}{6} = \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{6 π x}{6} = 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

\frac{6 π x}{6} = 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

\frac{6 π x}{6} : π x

\frac{6 π x}{6}

Divide the numbers:

\frac{6}{6} = 1

= π x

Simplify

6 \cdot \frac{π}{2} + 6 \cdot 2 πn : 3 π + 12 πn

6 \cdot \frac{π}{2} + 6 \cdot 2 πn

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 3 π + 12 πn

π x = 3 π + 12 πn

π x = 3 π + 12 πn

π x = 3 π + 12 πn

Divide both sides by

π

π x = 3 π + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{3 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} = \frac{3 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{3 π}{π} + \frac{12 πn}{π} : 3 + 12 n

\frac{3 π}{π} + \frac{12 πn}{π}

Cancel

\frac{3 π}{π} : 3

\frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 3 + 12 n

x = 3 + 12 n

x = 3 + 12 n

x = 3 + 12 n

Solve

\frac{π x}{6} = \frac{3 π}{2} + 2 πn : x = 9 + 12 n

\frac{π x}{6} = \frac{3 π}{2} + 2 πn

Multiply both sides by

6

\frac{π x}{6} = \frac{3 π}{2} + 2 πn

Multiply both sides by

6

\frac{6 π x}{6} = 6 \cdot \frac{3 π}{2} + 6 \cdot 2 πn

Simplify

\frac{6 π x}{6} = 6 \cdot \frac{3 π}{2} + 6 \cdot 2 πn

Simplify

\frac{6 π x}{6} : π x

\frac{6 π x}{6}

Divide the numbers:

\frac{6}{6} = 1

= π x

Simplify

6 \cdot \frac{3 π}{2} + 6 \cdot 2 πn : 9 π + 12 πn

6 \cdot \frac{3 π}{2} + 6 \cdot 2 πn

6 \cdot \frac{3 π}{2} = 9 π

6 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 6}{2}

Multiply the numbers:

3 \cdot 6 = 18

= \frac{18 π}{2}

Divide the numbers:

\frac{18}{2} = 9

= 9 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 9 π + 12 πn

π x = 9 π + 12 πn

π x = 9 π + 12 πn

π x = 9 π + 12 πn

Divide both sides by

π

π x = 9 π + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{9 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} = \frac{9 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{9 π}{π} + \frac{12 πn}{π} : 9 + 12 n

\frac{9 π}{π} + \frac{12 πn}{π}

Cancel

\frac{9 π}{π} : 9

\frac{9 π}{π}

Cancel the common factor:

π

= 9

= 9 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 9 + 12 n

x = 9 + 12 n

x = 9 + 12 n

x = 9 + 12 n

x = 3 + 12 n, x = 9 + 12 n

Solutions for the range

x

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for pi/3 cos((pix)/6)=0,x ?
The general solution for pi/3 cos((pix)/6)=0,x is No Solution for x\in\mathbb{R}