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

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

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cos(3x)=cos(2x)

Solution

cos (3 x) = cos (2 x)

Solution

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

Degrees

x = 0^{\circ} + 14 4^{\circ} n, x = 7 2^{\circ} + 14 4^{\circ} n

Solution steps

cos (3 x) = cos (2 x)

Subtract

cos (2 x)

from both sides

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

Rewrite using trig identities

- cos (2 x) + cos (3 x)

Use the Sum to Product identity:

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

= - 2 sin (\frac{3 x + 2 x}{2}) sin (\frac{3 x - 2 x}{2})

Simplify

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

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

Add similar elements:

3 x + 2 x = 5 x

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

Add similar elements:

3 x - 2 x = x

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

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

- 2 sin (\frac{5 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

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

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

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

General solutions for

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

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

\frac{5 x}{2} = 0 + 2 πn, \frac{5 x}{2} = π + 2 πn

\frac{5 x}{2} = 0 + 2 πn, \frac{5 x}{2} = π + 2 πn

Solve

\frac{5 x}{2} = 0 + 2 πn : x = \frac{4 πn}{5}

\frac{5 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{5 x}{2} = 2 πn

Multiply both sides by

2

\frac{5 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 5 x}{2} = 2 \cdot 2 πn

Simplify

5 x = 4 πn

5 x = 4 πn

Divide both sides by

5

5 x = 4 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{4 πn}{5}

Simplify

x = \frac{4 πn}{5}

x = \frac{4 πn}{5}

Solve

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

\frac{5 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = π + 2 πn

Multiply both sides by

2

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

Simplify

5 x = 2 π + 4 πn

5 x = 2 π + 4 πn

Divide both sides by

5

5 x = 2 π + 4 πn

Divide both sides by

5

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

Simplify

x = \frac{2 π}{5} + \frac{4 πn}{5}

x = \frac{2 π}{5} + \frac{4 πn}{5}

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

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

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

General solutions for

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

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

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

Solve

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplify

x = 4 πn

x = 4 πn

Solve

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

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

Combine all the solutions

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

Merge Overlapping Intervals

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

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Frequently Asked Questions (FAQ)

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