What is the general solution for cos(8x)-cos(4x)=0 ?

The general solution for cos(8x)-cos(4x)=0 is x=pin,x= pi/2+pin,x=(pin)/3 ,x= pi/6+(pin)/3

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cos(8x)-cos(4x)=0

Solution

cos (8 x) - cos (4 x) = 0

Solution

x = πn, x = \frac{π}{2} + πn, x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 6 0^{\circ} n, x = 3 0^{\circ} + 6 0^{\circ} n

Solution steps

cos (8 x) - cos (4 x) = 0

Rewrite using trig identities

- cos (4 x) + cos (8 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{8 x + 4 x}{2}) sin (\frac{8 x - 4 x}{2})

Simplify

- 2 sin (\frac{8 x + 4 x}{2}) sin (\frac{8 x - 4 x}{2}) : - 2 sin (6 x) sin (2 x)

- 2 sin (\frac{8 x + 4 x}{2}) sin (\frac{8 x - 4 x}{2})

\frac{8 x + 4 x}{2} = 6 x

\frac{8 x + 4 x}{2}

Add similar elements:

8 x + 4 x = 12 x

= \frac{12 x}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 x

= - 2 sin (6 x) sin (\frac{8 x - 4 x}{2})

\frac{8 x - 4 x}{2} = 2 x

\frac{8 x - 4 x}{2}

Add similar elements:

8 x - 4 x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= - 2 sin (6 x) sin (2 x)

= - 2 sin (6 x) sin (2 x)

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

Solving each part separately

sin (2 x) = 0 or sin (6 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 (6 x) = 0 : x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}

sin (6 x) = 0

General solutions for

sin (6 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}

6 x = 0 + 2 πn, 6 x = π + 2 πn

6 x = 0 + 2 πn, 6 x = π + 2 πn

Solve

6 x = 0 + 2 πn : x = \frac{πn}{3}

6 x = 0 + 2 πn

0 + 2 πn = 2 πn

6 x = 2 πn

Divide both sides by

6

6 x = 2 πn

Divide both sides by

6

\frac{6 x}{6} = \frac{2 πn}{6}

Simplify

x = \frac{πn}{3}

x = \frac{πn}{3}

Solve

6 x = π + 2 πn : x = \frac{π}{6} + \frac{πn}{3}

6 x = π + 2 πn

Divide both sides by

6

6 x = π + 2 πn

Divide both sides by

6

\frac{6 x}{6} = \frac{π}{6} + \frac{2 πn}{6}

Simplify

x = \frac{π}{6} + \frac{πn}{3}

x = \frac{π}{6} + \frac{πn}{3}

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}

Combine all the solutions

x = πn, x = \frac{π}{2} + πn, x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(8x)-cos(4x)=0 ?
The general solution for cos(8x)-cos(4x)=0 is x=pin,x= pi/2+pin,x=(pin)/3 ,x= pi/6+(pin)/3