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Popular >

cos(8x)-cos(2x)=sin(5x)

Solution

cos (8 x) - cos (2 x) = sin (5 x)

Solution

x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}, x = \frac{7 π}{18} + \frac{2 πn}{3}, x = \frac{11 π}{18} + \frac{2 πn}{3}

Degrees

x = 0^{\circ} + 7 2^{\circ} n, x = 3 6^{\circ} + 7 2^{\circ} n, x = 7 0^{\circ} + 12 0^{\circ} n, x = 11 0^{\circ} + 12 0^{\circ} n

Solution steps

cos (8 x) - cos (2 x) = sin (5 x)

Subtract

sin (5 x)

from both sides

cos (8 x) - cos (2 x) - sin (5 x) = 0

Rewrite using trig identities

- cos (2 x) + cos (8 x) - sin (5 x)

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

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

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

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

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

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

Add similar elements:

8 x + 2 x = 10 x

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

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

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

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

Add similar elements:

8 x - 2 x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

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

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

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

Factor

- sin (5 x) - 2 sin (3 x) sin (5 x) : - sin (5 x) (2 sin (3 x) + 1)

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

Factor out common term

- sin (5 x)

= - sin (5 x) (1 + 2 sin (3 x))

- sin (5 x) (2 sin (3 x) + 1) = 0

Solving each part separately

sin (5 x) = 0 or 2 sin (3 x) + 1 = 0

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}

2 sin (3 x) + 1 = 0 : x = \frac{7 π}{18} + \frac{2 πn}{3}, x = \frac{11 π}{18} + \frac{2 πn}{3}

2 sin (3 x) + 1 = 0

Move

1

to the right side

2 sin (3 x) + 1 = 0

Subtract

1

from both sides

2 sin (3 x) + 1 - 1 = 0 - 1

Simplify

2 sin (3 x) = - 1

2 sin (3 x) = - 1

Divide both sides by

2

2 sin (3 x) = - 1

Divide both sides by

2

\frac{2 sin ( 3 x )}{2} = \frac{- 1}{2}

Simplify

sin (3 x) = - \frac{1}{2}

sin (3 x) = - \frac{1}{2}

General solutions for

sin (3 x) = - \frac{1}{2}

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 = \frac{7 π}{6} + 2 πn, 3 x = \frac{11 π}{6} + 2 πn

3 x = \frac{7 π}{6} + 2 πn, 3 x = \frac{11 π}{6} + 2 πn

Solve

3 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{18} + \frac{2 πn}{3}

3 x = \frac{7 π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{7 π}{6} + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3} : \frac{7 π}{18} + \frac{2 πn}{3}

\frac{\frac{7 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{7 π}{6}}{3} = \frac{7 π}{18}

\frac{\frac{7 π}{6}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{7 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{7 π}{18}

= \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}

Solve

3 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{18} + \frac{2 πn}{3}

3 x = \frac{11 π}{6} + 2 πn

Divide both sides by

3

3 x = \frac{11 π}{6} + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{11 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{11 π}{6}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{11 π}{6}}{3} + \frac{2 πn}{3} : \frac{11 π}{18} + \frac{2 πn}{3}

\frac{\frac{11 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{11 π}{6}}{3} = \frac{11 π}{18}

\frac{\frac{11 π}{6}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{11 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{11 π}{18}

= \frac{11 π}{18} + \frac{2 πn}{3}

x = \frac{11 π}{18} + \frac{2 πn}{3}

x = \frac{11 π}{18} + \frac{2 πn}{3}

x = \frac{11 π}{18} + \frac{2 πn}{3}

x = \frac{7 π}{18} + \frac{2 πn}{3}, x = \frac{11 π}{18} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}, x = \frac{7 π}{18} + \frac{2 πn}{3}, x = \frac{11 π}{18} + \frac{2 πn}{3}

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