What is the general solution for cos(x+3pi)-cos(x)+1=0 ?

The general solution for cos(x+3pi)-cos(x)+1=0 is x=2pin-(3pi)/2+(7pi)/6 ,x=2pin-(3pi)/2+(11pi)/6

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cos(x+3pi)-cos(x)+1=0

Solution

cos (x + 3 π) - cos (x) + 1 = 0

Solution

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

Degrees

x = - 6 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x + 3 π) - cos (x) + 1 = 0

Rewrite using trig identities

cos (x + 3 π) - cos (x) + 1

Use the Sum to Product identity:

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

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

Simplify

1 - 2 sin (\frac{x + 3 π + x}{2}) sin (\frac{x + 3 π - x}{2}) : 1 + 2 sin (\frac{2 x + 3 π}{2})

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

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

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

x + 3 π + x = 2 x + 3 π

x + 3 π + x

Group like terms

= x + x + 3 π

Add similar elements:

x + x = 2 x

= 2 x + 3 π

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

x + 3 π - x = 3 π

x + 3 π - x

Group like terms

= x - x + 3 π

Add similar elements:

x - x = 0

= 3 π

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

sin (\frac{3 π}{2}) = - 1

sin (\frac{3 π}{2})

Rewrite using trig identities:

sin (π) cos (\frac{π}{2}) + cos (π) sin (\frac{π}{2})

sin (\frac{3 π}{2})

Write

sin (\frac{3 π}{2})

sin (π + \frac{π}{2})

= sin (π + \frac{π}{2})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{π}{2}) + cos (π) sin (\frac{π}{2})

= sin (π) cos (\frac{π}{2}) + cos (π) sin (\frac{π}{2})

Use the following trivial identity:

sin (π) = 0

sin (π)

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}

= 0

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{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}

= 1

= 0 \cdot 0 + (- 1) \cdot 1

Simplify

= - 1

= 2 (- 1) sin (\frac{2 x + 3 π}{2})

Refine

= - 2 sin (\frac{2 x + 3 π}{2})

= 1 - (- 2 sin (\frac{2 x + 3 π}{2}))

Apply rule

- (- a) = a

= 1 + 2 sin (\frac{2 x + 3 π}{2})

= 1 + 2 sin (\frac{2 x + 3 π}{2})

1 + 2 sin (\frac{2 x + 3 π}{2}) = 0

Move

1

to the right side

1 + 2 sin (\frac{2 x + 3 π}{2}) = 0

Subtract

1

from both sides

1 + 2 sin (\frac{2 x + 3 π}{2}) - 1 = 0 - 1

Simplify

2 sin (\frac{2 x + 3 π}{2}) = - 1

2 sin (\frac{2 x + 3 π}{2}) = - 1

Divide both sides by

2

2 sin (\frac{2 x + 3 π}{2}) = - 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

sin (\frac{2 x + 3 π}{2}) = - \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}

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 ( 2 x + 3 π )}{2} : 2 x + 3 π

\frac{2 ( 2 x + 3 π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x + 3 π

Simplify

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

2 x + 3 π = \frac{7 π}{3} + 4 πn

2 x + 3 π = \frac{7 π}{3} + 4 πn

2 x + 3 π = \frac{7 π}{3} + 4 πn

Move

3 π

to the right side

2 x + 3 π = \frac{7 π}{3} + 4 πn

Subtract

3 π

from both sides

2 x + 3 π - 3 π = \frac{7 π}{3} + 4 πn - 3 π

Simplify

2 x = \frac{7 π}{3} + 4 πn - 3 π

2 x = \frac{7 π}{3} + 4 πn - 3 π

Divide both sides by

2

2 x = \frac{7 π}{3} + 4 πn - 3 π

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{7 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{7 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{7 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2} : 2 πn - \frac{3 π}{2} + \frac{7 π}{6}

\frac{\frac{7 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2}

Group like terms

= - \frac{3 π}{2} + \frac{4 πn}{2} + \frac{\frac{7 π}{3}}{2}

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{7 π}{3}}{2} = \frac{7 π}{6}

\frac{\frac{7 π}{3}}{2}

Apply the fraction rule:

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

= \frac{7 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{7 π}{6}

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

Group like terms

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

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

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 ( 2 x + 3 π )}{2} : 2 x + 3 π

\frac{2 ( 2 x + 3 π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x + 3 π

Simplify

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 2}{6}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{6}

Cancel the common factor:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

2 x + 3 π = \frac{11 π}{3} + 4 πn

2 x + 3 π = \frac{11 π}{3} + 4 πn

2 x + 3 π = \frac{11 π}{3} + 4 πn

Move

3 π

to the right side

2 x + 3 π = \frac{11 π}{3} + 4 πn

Subtract

3 π

from both sides

2 x + 3 π - 3 π = \frac{11 π}{3} + 4 πn - 3 π

Simplify

2 x = \frac{11 π}{3} + 4 πn - 3 π

2 x = \frac{11 π}{3} + 4 πn - 3 π

Divide both sides by

2

2 x = \frac{11 π}{3} + 4 πn - 3 π

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{11 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{11 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{11 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2} : 2 πn - \frac{3 π}{2} + \frac{11 π}{6}

\frac{\frac{11 π}{3}}{2} + \frac{4 πn}{2} - \frac{3 π}{2}

Group like terms

= - \frac{3 π}{2} + \frac{4 πn}{2} + \frac{\frac{11 π}{3}}{2}

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{11 π}{3}}{2} = \frac{11 π}{6}

\frac{\frac{11 π}{3}}{2}

Apply the fraction rule:

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

= \frac{11 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{11 π}{6}

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

Group like terms

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

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

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x+3pi)-cos(x)+1=0 ?
The general solution for cos(x+3pi)-cos(x)+1=0 is x=2pin-(3pi)/2+(7pi)/6 ,x=2pin-(3pi)/2+(11pi)/6