What is the general solution for cos((2pix)/3)= 1/2 ?

The general solution for cos((2pix)/3)= 1/2 is x= 1/2+3n,x= 5/2+3n

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cos((2pix)/3)= 1/2

Solution

cos (\frac{2 π x}{3}) = \frac{1}{2}

Solution

x = \frac{1}{2} + 3 n, x = \frac{5}{2} + 3 n

Degrees

x = 28.64788 \dots^{\circ} + 171.88733 \dots^{\circ} n, x = 143.23944 \dots^{\circ} + 171.88733 \dots^{\circ} n

Solution steps

cos (\frac{2 π x}{3}) = \frac{1}{2}

General solutions for

cos (\frac{2 π x}{3}) = \frac{1}{2}

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 x}{3} = \frac{π}{3} + 2 πn, \frac{2 π x}{3} = \frac{5 π}{3} + 2 πn

\frac{2 π x}{3} = \frac{π}{3} + 2 πn, \frac{2 π x}{3} = \frac{5 π}{3} + 2 πn

Solve

\frac{2 π x}{3} = \frac{π}{3} + 2 πn : x = \frac{1}{2} + 3 n

\frac{2 π x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

\frac{2 π x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 \cdot 2 π x}{3} : 2 π x

\frac{3 \cdot 2 π x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π x

Simplify

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

3 \cdot \frac{π}{3} + 3 \cdot 2 πn

3 \cdot \frac{π}{3} = π

3 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 3}{3}

Cancel the common factor:

3

= π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= π + 6 πn

2 π x = π + 6 πn

2 π x = π + 6 πn

2 π x = π + 6 πn

Divide both sides by

2 π

2 π x = π + 6 πn

Divide both sides by

2 π

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

Simplify

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

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{π}{2 π} + \frac{6 πn}{2 π} : \frac{1}{2} + 3 n

\frac{π}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{π}{2 π} : \frac{1}{2}

\frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

= \frac{1}{2} + \frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 πn}{π}

Cancel the common factor:

π

= 3 n

= 3 n

= \frac{1}{2} + 3 n

x = \frac{1}{2} + 3 n

x = \frac{1}{2} + 3 n

x = \frac{1}{2} + 3 n

Solve

\frac{2 π x}{3} = \frac{5 π}{3} + 2 πn : x = \frac{5}{2} + 3 n

\frac{2 π x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{2 π x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{3 \cdot 2 π x}{3} = 3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π x}{3} = 3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π x}{3} : 2 π x

\frac{3 \cdot 2 π x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π x

Simplify

3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn : 5 π + 6 πn

3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{3} = 5 π

3 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 3}{3}

Cancel the common factor:

3

= 5 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 5 π + 6 πn

2 π x = 5 π + 6 πn

2 π x = 5 π + 6 πn

2 π x = 5 π + 6 πn

Divide both sides by

2 π

2 π x = 5 π + 6 πn

Divide both sides by

2 π

\frac{2 π x}{2 π} = \frac{5 π}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π x}{2 π} = \frac{5 π}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{5 π}{2 π} + \frac{6 πn}{2 π} : \frac{5}{2} + 3 n

\frac{5 π}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{5 π}{2 π} : \frac{5}{2}

\frac{5 π}{2 π}

Cancel the common factor:

π

= \frac{5}{2}

= \frac{5}{2} + \frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 πn}{π}

Cancel the common factor:

π

= 3 n

= 3 n

= \frac{5}{2} + 3 n

x = \frac{5}{2} + 3 n

x = \frac{5}{2} + 3 n

x = \frac{5}{2} + 3 n

x = \frac{1}{2} + 3 n, x = \frac{5}{2} + 3 n

Popular Examples

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- \frac{1}{2} = cos (2 x)

1/2 =cos(2x)

\frac{1}{2} = cos (2 x)

-sqrt(1+tan(x))=sec(x)

- 1 + tan (x) = sec (x)

sin(x/2)=sin(2x)

sin (\frac{x}{2}) = sin (2 x)

cos(x)=cos(1)

cos (x) = cos (1)

Frequently Asked Questions (FAQ)

What is the general solution for cos((2pix)/3)= 1/2 ?
The general solution for cos((2pix)/3)= 1/2 is x= 1/2+3n,x= 5/2+3n