What is the general solution for sin(2pi+x)-sin(2pi-x)=-1 ?

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

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sin(2pi+x)-sin(2pi-x)=-1

Solution

sin (2 π + x) - sin (2 π - x) = - 1

Solution

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

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (2 π + x) - sin (2 π - x) = - 1

Rewrite using trig identities

sin (2 π + x) - sin (2 π - x)

Use the Sum to Product identity:

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

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

Simplify

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

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

\frac{2 π + x - ( 2 π - x )}{2} = x

\frac{2 π + x - ( 2 π - x )}{2}

Expand

2 π + x - (2 π - x) : 2 x

2 π + x - (2 π - x)

- (2 π - x) : - 2 π + x

- (2 π - x)

Distribute parentheses

= - (2 π) - (- x)

Apply minus-plus rules

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

= - 2 π + x

= 2 π + x - 2 π + x

Simplify

2 π + x - 2 π + x : 2 x

2 π + x - 2 π + x

Group like terms

= x + x + 2 π - 2 π

Add similar elements:

x + x = 2 x

= 2 x + 2 π - 2 π

Add similar elements:

2 π - 2 π = 0

= 2 x

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

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

\frac{2 π + x + 2 π - x}{2} = 2 π

\frac{2 π + x + 2 π - x}{2}

2 π + x + 2 π - x = 4 π

2 π + x + 2 π - x

Group like terms

= x - x + 2 π + 2 π

Add similar elements:

x - x = 0

= 2 π + 2 π

Add similar elements:

2 π + 2 π = 4 π

= 4 π

= \frac{4 π}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

= 2 cos (2 π) sin (x)

cos (2 π) = 1

cos (2 π)

cos (2 π) = cos (0)

cos (2 π)

Rewrite

2 π

2 π + 0

= cos (2 π + 0)

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

cos (2 π + 0) = cos (0)

= cos (0)

= cos (0)

Use the following trivial identity:

cos (0) = 1

cos (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}

= 1

= 1

= 2 \cdot 1 \cdot sin (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 sin (x)

= 2 sin (x)

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

sin (x) = - \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}

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

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

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Frequently Asked Questions (FAQ)

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