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Popular Trigonometry >

cos(5x)=cos(5+x)

Solution

cos (5 x) = cos (5 + x)

Solution

x = πn + \frac{5}{4}, x = \frac{5}{4} + \frac{π}{2} + πn, x = \frac{2 πn}{3} - \frac{5}{6}, x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

Degrees

x = 71.61972 \dots^{\circ} + 18 0^{\circ} n, x = 161.61972 \dots^{\circ} + 18 0^{\circ} n, x = - 47.74648 \dots^{\circ} + 12 0^{\circ} n, x = 12.25351 \dots^{\circ} + 12 0^{\circ} n

Solution steps

cos (5 x) = cos (5 + x)

Subtract

cos (5 + x)

from both sides

cos (5 x) - cos (5 + x) = 0

Rewrite using trig identities

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

Simplify

- 2 sin (\frac{5 x + 5 + x}{2}) sin (\frac{5 x - ( 5 + x )}{2}) : - 2 sin (\frac{6 x + 5}{2}) sin (\frac{4 x - 5}{2})

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

5 x + 5 + x = 6 x + 5

5 x + 5 + x

Group like terms

= 5 x + x + 5

Add similar elements:

5 x + x = 6 x

= 6 x + 5

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

Expand

5 x - (5 + x) : 4 x - 5

5 x - (5 + x)

- (5 + x) : - 5 - x

- (5 + x)

Distribute parentheses

= - (5) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 5 - x

= 5 x - 5 - x

Simplify

5 x - 5 - x : 4 x - 5

5 x - 5 - x

Group like terms

= 5 x - x - 5

Add similar elements:

5 x - x = 4 x

= 4 x - 5

= 4 x - 5

= - 2 sin (\frac{6 x + 5}{2}) sin (\frac{4 x - 5}{2})

= - 2 sin (\frac{6 x + 5}{2}) sin (\frac{4 x - 5}{2})

- 2 sin (\frac{- 5 + 4 x}{2}) sin (\frac{5 + 6 x}{2}) = 0

Solving each part separately

sin (\frac{- 5 + 4 x}{2}) = 0 or sin (\frac{5 + 6 x}{2}) = 0

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

sin (\frac{- 5 + 4 x}{2}) = 0

General solutions for

sin (\frac{- 5 + 4 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 + 4 x}{2} = 0 + 2 πn, \frac{- 5 + 4 x}{2} = π + 2 πn

\frac{- 5 + 4 x}{2} = 0 + 2 πn, \frac{- 5 + 4 x}{2} = π + 2 πn

Solve

\frac{- 5 + 4 x}{2} = 0 + 2 πn : x = πn + \frac{5}{4}

\frac{- 5 + 4 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{- 5 + 4 x}{2} = 2 πn

Multiply both sides by

2

\frac{- 5 + 4 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( - 5 + 4 x )}{2} = 2 \cdot 2 πn

Simplify

- 5 + 4 x = 4 πn

- 5 + 4 x = 4 πn

Move

5

to the right side

- 5 + 4 x = 4 πn

Add

5

to both sides

- 5 + 4 x + 5 = 4 πn + 5

Simplify

4 x = 4 πn + 5

4 x = 4 πn + 5

Divide both sides by

4

4 x = 4 πn + 5

Divide both sides by

4

\frac{4 x}{4} = \frac{4 πn}{4} + \frac{5}{4}

Simplify

x = πn + \frac{5}{4}

x = πn + \frac{5}{4}

Solve

\frac{- 5 + 4 x}{2} = π + 2 πn : x = \frac{5}{4} + \frac{π}{2} + πn

\frac{- 5 + 4 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{- 5 + 4 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( - 5 + 4 x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

- 5 + 4 x = 2 π + 4 πn

- 5 + 4 x = 2 π + 4 πn

Move

5

to the right side

- 5 + 4 x = 2 π + 4 πn

Add

5

to both sides

- 5 + 4 x + 5 = 2 π + 4 πn + 5

Simplify

4 x = 2 π + 4 πn + 5

4 x = 2 π + 4 πn + 5

Divide both sides by

4

4 x = 2 π + 4 πn + 5

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{2 π}{4} + \frac{4 πn}{4} + \frac{5}{4} : \frac{5}{4} + \frac{π}{2} + πn

\frac{2 π}{4} + \frac{4 πn}{4} + \frac{5}{4}

Group like terms

= \frac{5}{4} + \frac{2 π}{4} + \frac{4 πn}{4}

Cancel

\frac{2 π}{4} : \frac{π}{2}

\frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{5}{4} + \frac{π}{2} + \frac{4 πn}{4}

Divide the numbers:

\frac{4}{4} = 1

= \frac{5}{4} + \frac{π}{2} + πn

x = \frac{5}{4} + \frac{π}{2} + πn

x = \frac{5}{4} + \frac{π}{2} + πn

x = \frac{5}{4} + \frac{π}{2} + πn

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

sin (\frac{5 + 6 x}{2}) = 0 : x = \frac{2 πn}{3} - \frac{5}{6}, x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

sin (\frac{5 + 6 x}{2}) = 0

General solutions for

sin (\frac{5 + 6 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 + 6 x}{2} = 0 + 2 πn, \frac{5 + 6 x}{2} = π + 2 πn

\frac{5 + 6 x}{2} = 0 + 2 πn, \frac{5 + 6 x}{2} = π + 2 πn

Solve

\frac{5 + 6 x}{2} = 0 + 2 πn : x = \frac{2 πn}{3} - \frac{5}{6}

\frac{5 + 6 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{5 + 6 x}{2} = 2 πn

Multiply both sides by

2

\frac{5 + 6 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 5 + 6 x )}{2} = 2 \cdot 2 πn

Simplify

5 + 6 x = 4 πn

5 + 6 x = 4 πn

Move

5

to the right side

5 + 6 x = 4 πn

Subtract

5

from both sides

5 + 6 x - 5 = 4 πn - 5

Simplify

6 x = 4 πn - 5

6 x = 4 πn - 5

Divide both sides by

6

6 x = 4 πn - 5

Divide both sides by

6

\frac{6 x}{6} = \frac{4 πn}{6} - \frac{5}{6}

Simplify

x = \frac{2 πn}{3} - \frac{5}{6}

x = \frac{2 πn}{3} - \frac{5}{6}

Solve

\frac{5 + 6 x}{2} = π + 2 πn : x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

\frac{5 + 6 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{5 + 6 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 5 + 6 x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

5 + 6 x = 2 π + 4 πn

5 + 6 x = 2 π + 4 πn

Move

5

to the right side

5 + 6 x = 2 π + 4 πn

Subtract

5

from both sides

5 + 6 x - 5 = 2 π + 4 πn - 5

Simplify

6 x = 2 π + 4 πn - 5

6 x = 2 π + 4 πn - 5

Divide both sides by

6

6 x = 2 π + 4 πn - 5

Divide both sides by

6

\frac{6 x}{6} = \frac{2 π}{6} + \frac{4 πn}{6} - \frac{5}{6}

Simplify

\frac{6 x}{6} = \frac{2 π}{6} + \frac{4 πn}{6} - \frac{5}{6}

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{2 π}{6} + \frac{4 πn}{6} - \frac{5}{6} : - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

\frac{2 π}{6} + \frac{4 πn}{6} - \frac{5}{6}

Group like terms

= - \frac{5}{6} + \frac{2 π}{6} + \frac{4 πn}{6}

Cancel

\frac{2 π}{6} : \frac{π}{3}

\frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= - \frac{5}{6} + \frac{π}{3} + \frac{4 πn}{6}

Cancel

\frac{4 πn}{6} : \frac{2 πn}{3}

\frac{4 πn}{6}

Cancel the common factor:

2

= \frac{2 πn}{3}

= - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

x = \frac{2 πn}{3} - \frac{5}{6}, x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

Combine all the solutions

x = πn + \frac{5}{4}, x = \frac{5}{4} + \frac{π}{2} + πn, x = \frac{2 πn}{3} - \frac{5}{6}, x = - \frac{5}{6} + \frac{π}{3} + \frac{2 πn}{3}

Graph

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