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Popular >

0=-cos(8x)+cos(2x)

Solution

0 = - cos (8 x) + cos (2 x)

Solution

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}

Degrees

x = 0^{\circ} + 12 0^{\circ} n, x = 6 0^{\circ} + 12 0^{\circ} n, x = 0^{\circ} + 7 2^{\circ} n, x = 3 6^{\circ} + 7 2^{\circ} n

Solution steps

0 = - cos (8 x) + cos (2 x)

Switch sides

- cos (8 x) + cos (2 x) = 0

Rewrite using trig identities

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

Simplify

- 2 sin (\frac{2 x + 8 x}{2}) sin (\frac{2 x - 8 x}{2}) : 2 sin (3 x) sin (5 x)

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

\frac{2 x + 8 x}{2} = 5 x

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

Add similar elements:

2 x + 8 x = 10 x

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

= - 2 sin (5 x) sin (\frac{2 x - 8 x}{2})

\frac{2 x - 8 x}{2} = - 3 x

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

Add similar elements:

2 x - 8 x = - 6 x

= \frac{- 6 x}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3 x

= - 2 sin (5 x) sin (- 3 x)

Use the negative angle identity:

sin (- x) = - sin (x)

= - 2 (- sin (3 x)) sin (5 x)

Apply rule

- (- a) = a

= 2 sin (3 x) sin (5 x)

= 2 sin (3 x) sin (5 x)

2 sin (3 x) sin (5 x) = 0

Solving each part separately

sin (3 x) = 0 or sin (5 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 (5 x) = 0 : x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}

sin (5 x) = 0

General solutions for

sin (5 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}

5 x = 0 + 2 πn, 5 x = π + 2 πn

5 x = 0 + 2 πn, 5 x = π + 2 πn

Solve

5 x = 0 + 2 πn : x = \frac{2 πn}{5}

5 x = 0 + 2 πn

0 + 2 πn = 2 πn

5 x = 2 πn

Divide both sides by

5

5 x = 2 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{2 πn}{5}

Simplify

x = \frac{2 πn}{5}

x = \frac{2 πn}{5}

Solve

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

5 x = π + 2 πn

Divide both sides by

5

5 x = π + 2 πn

Divide both sides by

5

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

Simplify

x = \frac{π}{5} + \frac{2 πn}{5}

x = \frac{π}{5} + \frac{2 πn}{5}

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

Combine all the solutions

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}

Graph

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