What is the general solution for cos(2x)+cos(x)=0 ?

The general solution for cos(2x)+cos(x)=0 is x= pi/3+(4pin)/3 ,x=pi+(4pin)/3

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cos(2x)+cos(x)=0

Solution

cos (2 x) + cos (x) = 0

Solution

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

Degrees

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n

Solution steps

cos (2 x) + cos (x) = 0

Rewrite using trig identities

cos (2 x) + cos (x)

Use the Sum to Product identity:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

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

Simplify

2 cos (\frac{2 x + x}{2}) cos (\frac{2 x - x}{2}) : 2 cos (\frac{3 x}{2}) cos (\frac{x}{2})

2 cos (\frac{2 x + x}{2}) cos (\frac{2 x - x}{2})

Add similar elements:

2 x + x = 3 x

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

Add similar elements:

2 x - x = x

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

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

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

Solving each part separately

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

cos (\frac{3 x}{2}) = 0 : x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

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

General solutions for

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

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

Solve

\frac{3 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{3} + \frac{4 πn}{3}

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

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

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

Divide both sides by

3

3 x = π + 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{π}{3} + \frac{4 πn}{3}

Simplify

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

Solve

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn : x = π + \frac{4 πn}{3}

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

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

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

Divide both sides by

3

3 x = 3 π + 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{3 π}{3} + \frac{4 πn}{3}

Simplify

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

cos (\frac{x}{2}) = 0 : x = π + 4 πn, x = 3 π + 4 πn

cos (\frac{x}{2}) = 0

General solutions for

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

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

Solve

\frac{x}{2} = \frac{π}{2} + 2 πn : x = π + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

Solve

\frac{x}{2} = \frac{3 π}{2} + 2 πn : x = 3 π + 4 πn

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = π + 4 πn, x = 3 π + 4 πn

Combine all the solutions

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}, x = π + 4 πn, x = 3 π + 4 πn

Merge Overlapping Intervals

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)+cos(x)=0 ?
The general solution for cos(2x)+cos(x)=0 is x= pi/3+(4pin)/3 ,x=pi+(4pin)/3