What is the general solution for cos(6x)=cos(2x) ?

The general solution for cos(6x)=cos(2x) is x=pin,x= pi/2+pin,x=(pin)/2 ,x= pi/4+(pin)/2

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cos(6x)=cos(2x)

Solution

cos (6 x) = cos (2 x)

Solution

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

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 9 0^{\circ} n, x = 4 5^{\circ} + 9 0^{\circ} n

Solution steps

cos (6 x) = cos (2 x)

Subtract

cos (2 x)

from both sides

cos (6 x) - cos (2 x) = 0

Rewrite using trig identities

- cos (2 x) + cos (6 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{6 x + 2 x}{2}) sin (\frac{6 x - 2 x}{2})

Simplify

- 2 sin (\frac{6 x + 2 x}{2}) sin (\frac{6 x - 2 x}{2}) : - 2 sin (4 x) sin (2 x)

- 2 sin (\frac{6 x + 2 x}{2}) sin (\frac{6 x - 2 x}{2})

\frac{6 x + 2 x}{2} = 4 x

\frac{6 x + 2 x}{2}

Add similar elements:

6 x + 2 x = 8 x

= \frac{8 x}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 x

= - 2 sin (4 x) sin (\frac{6 x - 2 x}{2})

\frac{6 x - 2 x}{2} = 2 x

\frac{6 x - 2 x}{2}

Add similar elements:

6 x - 2 x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= - 2 sin (4 x) sin (2 x)

= - 2 sin (4 x) sin (2 x)

- 2 sin (2 x) sin (4 x) = 0

Solving each part separately

sin (2 x) = 0 or sin (4 x) = 0

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

sin (2 x) = 0

General solutions for

sin (2 x) = 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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

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

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

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

sin (4 x) = 0

General solutions for

sin (4 x) = 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}

4 x = 0 + 2 πn, 4 x = π + 2 πn

4 x = 0 + 2 πn, 4 x = π + 2 πn

Solve

4 x = 0 + 2 πn : x = \frac{πn}{2}

4 x = 0 + 2 πn

0 + 2 πn = 2 πn

4 x = 2 πn

Divide both sides by

4

4 x = 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4}

Simplify

x = \frac{πn}{2}

x = \frac{πn}{2}

Solve

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

4 x = π + 2 πn

Divide both sides by

4

4 x = π + 2 πn

Divide both sides by

4

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

Simplify

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{πn}{2}

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

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos(6x)=cos(2x) ?
The general solution for cos(6x)=cos(2x) is x=pin,x= pi/2+pin,x=(pin)/2 ,x= pi/4+(pin)/2