What is the general solution for sin(x)=sin(x+1) ?

The general solution for sin(x)=sin(x+1) is x= pi/2-1/2+2pin,x=-1/2+(3pi)/2+2pin

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sin(x)=sin(x+1)

Solution

sin (x) = sin (x + 1)

Solution

x = \frac{π}{2} - \frac{1}{2} + 2 πn, x = - \frac{1}{2} + \frac{3 π}{2} + 2 πn

Degrees

x = 61.35211 \dots^{\circ} + 36 0^{\circ} n, x = 241.35211 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) = sin (x + 1)

Subtract

sin (x + 1)

from both sides

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

Rewrite using trig identities

sin (x) - sin (x + 1)

Use the Sum to Product identity:

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

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

Simplify

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

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

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

\frac{x - ( x + 1 )}{2}

Expand

x - (x + 1) : - 1

x - (x + 1)

- (x + 1) : - x - 1

- (x + 1)

Distribute parentheses

= - (x) - (1)

Apply minus-plus rules

+ (- a) = - a

= - x - 1

= x - x - 1

Add similar elements:

x - x = 0

= - 1

= \frac{- 1}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

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

Simplify

sin (- \frac{1}{2}) : - sin (\frac{1}{2})

sin (- \frac{1}{2})

Use the following property:

sin (- x) = - sin (x)

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

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

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

Add similar elements:

x + x = 2 x

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

Remove parentheses:

(- a) = - a

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

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

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

Divide both sides by

- 2 sin (\frac{1}{2})

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

Divide both sides by

- 2 sin (\frac{1}{2})

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

Simplify

cos (\frac{2 x + 1}{2}) = 0

cos (\frac{2 x + 1}{2}) = 0

General solutions for

cos (\frac{2 x + 1}{2}) = 0

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}

\frac{2 x + 1}{2} = \frac{π}{2} + 2 πn, \frac{2 x + 1}{2} = \frac{3 π}{2} + 2 πn

\frac{2 x + 1}{2} = \frac{π}{2} + 2 πn, \frac{2 x + 1}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{2 x + 1}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{2} - \frac{1}{2} + 2 πn

\frac{2 x + 1}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 x + 1}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( 2 x + 1 )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x + 1 )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x + 1 )}{2} : 2 x + 1

\frac{2 ( 2 x + 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x + 1

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

2 x + 1 = π + 4 πn

2 x + 1 = π + 4 πn

2 x + 1 = π + 4 πn

Move

1

to the right side

2 x + 1 = π + 4 πn

Subtract

1

from both sides

2 x + 1 - 1 = π + 4 πn - 1

Simplify

2 x = π + 4 πn - 1

2 x = π + 4 πn - 1

Divide both sides by

2

2 x = π + 4 πn - 1

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{4 πn}{2} - \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{π}{2} + \frac{4 πn}{2} - \frac{1}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} + \frac{4 πn}{2} - \frac{1}{2} : \frac{π}{2} - \frac{1}{2} + 2 πn

\frac{π}{2} + \frac{4 πn}{2} - \frac{1}{2}

Group like terms

= \frac{π}{2} - \frac{1}{2} + \frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= \frac{π}{2} - \frac{1}{2} + 2 πn

x = \frac{π}{2} - \frac{1}{2} + 2 πn

x = \frac{π}{2} - \frac{1}{2} + 2 πn

x = \frac{π}{2} - \frac{1}{2} + 2 πn

Solve

\frac{2 x + 1}{2} = \frac{3 π}{2} + 2 πn : x = - \frac{1}{2} + \frac{3 π}{2} + 2 πn

\frac{2 x + 1}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 x + 1}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( 2 x + 1 )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x + 1 )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x + 1 )}{2} : 2 x + 1

\frac{2 ( 2 x + 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x + 1

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

2 x + 1 = 3 π + 4 πn

2 x + 1 = 3 π + 4 πn

2 x + 1 = 3 π + 4 πn

Move

1

to the right side

2 x + 1 = 3 π + 4 πn

Subtract

1

from both sides

2 x + 1 - 1 = 3 π + 4 πn - 1

Simplify

2 x = 3 π + 4 πn - 1

2 x = 3 π + 4 πn - 1

Divide both sides by

2

2 x = 3 π + 4 πn - 1

Divide both sides by

2

\frac{2 x}{2} = \frac{3 π}{2} + \frac{4 πn}{2} - \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{3 π}{2} + \frac{4 πn}{2} - \frac{1}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{3 π}{2} + \frac{4 πn}{2} - \frac{1}{2} : - \frac{1}{2} + \frac{3 π}{2} + 2 πn

\frac{3 π}{2} + \frac{4 πn}{2} - \frac{1}{2}

Group like terms

= - \frac{1}{2} + \frac{3 π}{2} + \frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= - \frac{1}{2} + \frac{3 π}{2} + 2 πn

x = - \frac{1}{2} + \frac{3 π}{2} + 2 πn

x = - \frac{1}{2} + \frac{3 π}{2} + 2 πn

x = - \frac{1}{2} + \frac{3 π}{2} + 2 πn

x = \frac{π}{2} - \frac{1}{2} + 2 πn, x = - \frac{1}{2} + \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=sin(x+1) ?
The general solution for sin(x)=sin(x+1) is x= pi/2-1/2+2pin,x=-1/2+(3pi)/2+2pin