What is the general solution for cos(2x)+cos(3x)=0 ?

The general solution for cos(2x)+cos(3x)=0 is x= pi/5+(4pin)/5 ,x=(3pi)/5+(4pin)/5

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cos(2x)+cos(3x)=0

Solution

cos (2 x) + cos (3 x) = 0

Solution

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

Degrees

x = 3 6^{\circ} + 14 4^{\circ} n, x = 10 8^{\circ} + 14 4^{\circ} n

Solution steps

cos (2 x) + cos (3 x) = 0

Rewrite using trig identities

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

Simplify

2 cos (\frac{2 x + 3 x}{2}) cos (\frac{2 x - 3 x}{2}) : 2 cos (\frac{x}{2}) cos (\frac{5 x}{2})

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

Add similar elements:

2 x + 3 x = 5 x

= 2 cos (\frac{5 x}{2}) cos (\frac{2 x - 3 x}{2})

\frac{2 x - 3 x}{2} = - \frac{x}{2}

\frac{2 x - 3 x}{2}

Add similar elements:

2 x - 3 x = - x

= \frac{- x}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{x}{2}

= 2 cos (\frac{5 x}{2}) cos (- \frac{x}{2})

Use the negative angle identity:

cos (- x) = cos (x)

= 2 cos (\frac{x}{2}) cos (\frac{5 x}{2})

= 2 cos (\frac{x}{2}) cos (\frac{5 x}{2})

2 cos (\frac{5 x}{2}) cos (\frac{x}{2}) = 0

Solving each part separately

cos (\frac{5 x}{2}) = 0 or cos (\frac{x}{2}) = 0

cos (\frac{5 x}{2}) = 0 : x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

cos (\frac{5 x}{2}) = 0

General solutions for

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

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

Solve

\frac{5 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{5} + \frac{4 πn}{5}

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 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

5 x = π + 4 πn

5 x = π + 4 πn

5 x = π + 4 πn

Divide both sides by

5

5 x = π + 4 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5}

Simplify

x = \frac{π}{5} + \frac{4 πn}{5}

x = \frac{π}{5} + \frac{4 πn}{5}

Solve

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{5} + \frac{4 πn}{5}

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 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

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

Divide both sides by

5

5 x = 3 π + 4 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{3 π}{5} + \frac{4 πn}{5}

Simplify

x = \frac{3 π}{5} + \frac{4 πn}{5}

x = \frac{3 π}{5} + \frac{4 πn}{5}

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

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}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = π + 4 πn, x = 3 π + 4 πn

Merge Overlapping Intervals

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)+cos(3x)=0 ?
The general solution for cos(2x)+cos(3x)=0 is x= pi/5+(4pin)/5 ,x=(3pi)/5+(4pin)/5