What is the general solution for cos(4x)=cos(2x) ?

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

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Popular >

cos(4x)=cos(2x)

Solution

cos (4 x) = cos (2 x)

Solution

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

Degrees

x = 0^{\circ} + 12 0^{\circ} n, x = 6 0^{\circ} + 12 0^{\circ} n

Solution steps

cos (4 x) = cos (2 x)

Subtract

cos (2 x)

from both sides

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

Rewrite using trig identities

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

Simplify

- 2 sin (\frac{4 x + 2 x}{2}) sin (\frac{4 x - 2 x}{2}) : - 2 sin (3 x) sin (x)

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

\frac{4 x + 2 x}{2} = 3 x

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

Add similar elements:

4 x + 2 x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

= - 2 sin (3 x) sin (\frac{4 x - 2 x}{2})

\frac{4 x - 2 x}{2} = x

\frac{4 x - 2 x}{2}

Add similar elements:

4 x - 2 x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= - 2 sin (3 x) sin (x)

= - 2 sin (3 x) sin (x)

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

Solving each part separately

sin (3 x) = 0 or sin (x) = 0

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

sin (3 x) = 0

General solutions for

sin (3 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}

3 x = 0 + 2 πn, 3 x = π + 2 πn

3 x = 0 + 2 πn, 3 x = π + 2 πn

Solve

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

Divide both sides by

3

3 x = 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3}

Simplify

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

Solve

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

3 x = π + 2 πn

Divide both sides by

3

3 x = π + 2 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combine all the solutions

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

Merge Overlapping Intervals

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

Popular Examples

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

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

10sin(x)tan(x)-9tan(x)=0

10 sin (x) tan (x) - 9 tan (x) = 0

-sin(t)=0

- sin (t) = 0

cos(x)=4

cos (x) = 4

tan(x)= 5/6

tan (x) = \frac{5}{6}

Frequently Asked Questions (FAQ)

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