What is the general solution for cos(x)=cos(x-c) ?

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

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cos(x)=cos(x-c)

Solution

cos (x) = cos (x - c)

Solution

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

Solution steps

cos (x) = cos (x - c)

Subtract

cos (x - c)

from both sides

cos (x) - cos (x - c) = 0

Rewrite using trig identities

cos (x) - cos (x - c)

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

Simplify

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

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

Add similar elements:

x + x = 2 x

= - 2 sin (\frac{2 x - c}{2}) sin (\frac{x - ( x - c )}{2})

Expand

x - (x - c) : c

x - (x - c)

- (x - c) : - x + c

- (x - c)

Distribute parentheses

= - (x) - (- c)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - x + c

= x - x + c

Add similar elements:

x - x = 0

= c

= - 2 sin (\frac{c}{2}) sin (\frac{2 x - c}{2})

= - 2 sin (\frac{c}{2}) sin (\frac{2 x - c}{2})

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

General solutions for

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

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

Solve

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

\frac{2 x - c}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{2 x - c}{2} = 2 πn

Multiply both sides by

2

\frac{2 x - c}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 2 x - c )}{2} = 2 \cdot 2 πn

Simplify

2 x - c = 4 πn

2 x - c = 4 πn

Move

c

to the right side

2 x - c = 4 πn

Add

c

to both sides

2 x - c + c = 4 πn + c

Simplify

2 x = 4 πn + c

2 x = 4 πn + c

Divide both sides by

2

2 x = 4 πn + c

Divide both sides by

2

\frac{2 x}{2} = \frac{4 πn}{2} + \frac{c}{2}

Simplify

x = 2 πn + \frac{c}{2}

x = 2 πn + \frac{c}{2}

Solve

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

\frac{2 x - c}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x - c}{2} = π + 2 πn

Multiply both sides by

2

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

Simplify

2 x - c = 2 π + 4 πn

2 x - c = 2 π + 4 πn

Move

c

to the right side

2 x - c = 2 π + 4 πn

Add

c

to both sides

2 x - c + c = 2 π + 4 πn + c

Simplify

2 x = 2 π + 4 πn + c

2 x = 2 π + 4 πn + c

Divide both sides by

2

2 x = 2 π + 4 πn + c

Divide both sides by

2

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

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

= π + \frac{4 πn}{2} + \frac{c}{2}

Divide the numbers:

\frac{4}{2} = 2

= π + 2 πn + \frac{c}{2}

x = π + 2 πn + \frac{c}{2}

x = π + 2 πn + \frac{c}{2}

x = π + 2 πn + \frac{c}{2}

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

Popular Examples

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Frequently Asked Questions (FAQ)

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