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Popular >

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

Solution

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

Solution

x = \frac{4 πn}{5}, x = \frac{2 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{7}, x = \frac{2 π}{7} + \frac{4 πn}{7}

Degrees

x = 0^{\circ} + 14 4^{\circ} n, x = 7 2^{\circ} + 14 4^{\circ} n, x = 0^{\circ} + 102.85714 \dots^{\circ} n, x = 51.42857 \dots^{\circ} + 102.85714 \dots^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Simplify

- 2 sin (\frac{6 x + x}{2}) sin (\frac{6 x - x}{2}) : - 2 sin (\frac{7 x}{2}) sin (\frac{5 x}{2})

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

Add similar elements:

6 x + x = 7 x

= - 2 sin (\frac{7 x}{2}) sin (\frac{6 x - x}{2})

Add similar elements:

6 x - x = 5 x

= - 2 sin (\frac{7 x}{2}) sin (\frac{5 x}{2})

= - 2 sin (\frac{7 x}{2}) sin (\frac{5 x}{2})

- 2 sin (\frac{5 x}{2}) sin (\frac{7 x}{2}) = 0

Solving each part separately

sin (\frac{5 x}{2}) = 0 or sin (\frac{7 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{7 x}{2}) = 0 : x = \frac{4 πn}{7}, x = \frac{2 π}{7} + \frac{4 πn}{7}

sin (\frac{7 x}{2}) = 0

General solutions for

sin (\frac{7 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{7 x}{2} = 0 + 2 πn, \frac{7 x}{2} = π + 2 πn

\frac{7 x}{2} = 0 + 2 πn, \frac{7 x}{2} = π + 2 πn

Solve

\frac{7 x}{2} = 0 + 2 πn : x = \frac{4 πn}{7}

\frac{7 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{7 x}{2} = 2 πn

Multiply both sides by

2

\frac{7 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 7 x}{2} = 2 \cdot 2 πn

Simplify

7 x = 4 πn

7 x = 4 πn

Divide both sides by

7

7 x = 4 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{4 πn}{7}

Simplify

x = \frac{4 πn}{7}

x = \frac{4 πn}{7}

Solve

\frac{7 x}{2} = π + 2 πn : x = \frac{2 π}{7} + \frac{4 πn}{7}

\frac{7 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{7 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 \cdot 7 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

7 x = 2 π + 4 πn

7 x = 2 π + 4 πn

Divide both sides by

7

7 x = 2 π + 4 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{2 π}{7} + \frac{4 πn}{7}

Simplify

x = \frac{2 π}{7} + \frac{4 πn}{7}

x = \frac{2 π}{7} + \frac{4 πn}{7}

x = \frac{4 πn}{7}, x = \frac{2 π}{7} + \frac{4 πn}{7}

Combine all the solutions

x = \frac{4 πn}{5}, x = \frac{2 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{7}, x = \frac{2 π}{7} + \frac{4 πn}{7}

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