What is the general solution for cos(x+3)=cos(x) ?

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

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cos(x+3)=cos(x)

Solution

cos (x + 3) = cos (x)

Solution

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

Degrees

x = - 85.94366 \dots^{\circ} + 36 0^{\circ} n, x = 94.05633 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos (x + 3) = cos (x)

Subtract

cos (x)

from both sides

cos (x + 3) - cos (x) = 0

Rewrite using trig identities

cos (x + 3) - 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{x + 3 + x}{2}) sin (\frac{x + 3 - x}{2})

Simplify

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

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

x + 3 + x = 2 x + 3

x + 3 + x

Group like terms

= x + x + 3

Add similar elements:

x + x = 2 x

= 2 x + 3

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

x + 3 - x = 3

x + 3 - x

Group like terms

= x - x + 3

Add similar elements:

x - x = 0

= 3

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

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

- 2 sin (\frac{3}{2}) sin (\frac{2 x + 3}{2}) = 0

Divide both sides by

- 2 sin (\frac{3}{2})

- 2 sin (\frac{3}{2}) sin (\frac{2 x + 3}{2}) = 0

Divide both sides by

- 2 sin (\frac{3}{2})

\frac{- 2 sin ( \frac{3}{2} ) sin ( \frac{2 x + 3}{2} )}{- 2 sin ( \frac{3}{2} )} = \frac{0}{- 2 sin ( \frac{3}{2} )}

Simplify

sin (\frac{2 x + 3}{2}) = 0

sin (\frac{2 x + 3}{2}) = 0

General solutions for

sin (\frac{2 x + 3}{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{2 x + 3}{2} = 0 + 2 πn, \frac{2 x + 3}{2} = π + 2 πn

\frac{2 x + 3}{2} = 0 + 2 πn, \frac{2 x + 3}{2} = π + 2 πn

Solve

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

\frac{2 x + 3}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{2 x + 3}{2} = 2 πn

Multiply both sides by

2

\frac{2 x + 3}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 2 x + 3 )}{2} = 2 \cdot 2 πn

Simplify

2 x + 3 = 4 πn

2 x + 3 = 4 πn

Move

3

to the right side

2 x + 3 = 4 πn

Subtract

3

from both sides

2 x + 3 - 3 = 4 πn - 3

Simplify

2 x = 4 πn - 3

2 x = 4 πn - 3

Divide both sides by

2

2 x = 4 πn - 3

Divide both sides by

2

\frac{2 x}{2} = \frac{4 πn}{2} - \frac{3}{2}

Simplify

x = 2 πn - \frac{3}{2}

x = 2 πn - \frac{3}{2}

Solve

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

\frac{2 x + 3}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x + 3}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 2 x + 3 )}{2} = 2 π + 2 \cdot 2 πn

Simplify

2 x + 3 = 2 π + 4 πn

2 x + 3 = 2 π + 4 πn

Move

3

to the right side

2 x + 3 = 2 π + 4 πn

Subtract

3

from both sides

2 x + 3 - 3 = 2 π + 4 πn - 3

Simplify

2 x = 2 π + 4 πn - 3

2 x = 2 π + 4 πn - 3

Divide both sides by

2

2 x = 2 π + 4 πn - 3

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{2 π}{2} + \frac{4 πn}{2} - \frac{3}{2}

Group like terms

= - \frac{3}{2} + \frac{2 π}{2} + \frac{4 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{3}{2} + π + \frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= - \frac{3}{2} + π + 2 πn

x = - \frac{3}{2} + π + 2 πn

x = - \frac{3}{2} + π + 2 πn

x = - \frac{3}{2} + π + 2 πn

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

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Frequently Asked Questions (FAQ)

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