What is the general solution for 2cos((2x)/3)-1=0 ?

The general solution for 2cos((2x)/3)-1=0 is x= pi/2+3pin,x=(5pi)/2+3pin

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2cos((2x)/3)-1=0

Solution

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

Solution

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

Degrees

x = 9 0^{\circ} + 54 0^{\circ} n, x = 45 0^{\circ} + 54 0^{\circ} n

Solution steps

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

2 cos (\frac{2 x}{3}) = 1

2 cos (\frac{2 x}{3}) = 1

Divide both sides by

2

2 cos (\frac{2 x}{3}) = 1

Divide both sides by

2

\frac{2 cos ( \frac{2 x}{3} )}{2} = \frac{1}{2}

Simplify

cos (\frac{2 x}{3}) = \frac{1}{2}

cos (\frac{2 x}{3}) = \frac{1}{2}

General solutions for

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

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

Solve

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

\frac{2 x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

\frac{2 x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 \cdot 2 x}{3} : 2 x

\frac{3 \cdot 2 x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 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 x = π + 6 πn

2 x = π + 6 πn

2 x = π + 6 πn

Divide both sides by

2

2 x = π + 6 πn

Divide both sides by

2

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

Simplify

x = \frac{π}{2} + 3 πn

x = \frac{π}{2} + 3 πn

Solve

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

\frac{2 x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{2 x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 \cdot 2 x}{3} : 2 x

\frac{3 \cdot 2 x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 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 x = 5 π + 6 πn

2 x = 5 π + 6 πn

2 x = 5 π + 6 πn

Divide both sides by

2

2 x = 5 π + 6 πn

Divide both sides by

2

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

Simplify

x = \frac{5 π}{2} + 3 πn

x = \frac{5 π}{2} + 3 πn

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

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos((2x)/3)-1=0 ?
The general solution for 2cos((2x)/3)-1=0 is x= pi/2+3pin,x=(5pi)/2+3pin