What is the general solution for sin(x+pi)+sin(x-pi)=0 ?

The general solution for sin(x+pi)+sin(x-pi)=0 is x=2pin,x=pi+2pin

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

Solution

sin (x + π) + sin (x - π) = 0

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x + π) + sin (x - π) = 0

Rewrite using trig identities

sin (x + π) + sin (x - π)

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

Simplify

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

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

\frac{x + π + x - π}{2} = x

\frac{x + π + x - π}{2}

x + π + x - π = 2 x

x + π + x - π

Group like terms

= x + x + π - π

Add similar elements:

x + x = 2 x

= 2 x + π - π

Add similar elements:

π - π = 0

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

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

\frac{x + π - ( x - π )}{2} = π

\frac{x + π - ( x - π )}{2}

Expand

x + π - (x - π) : 2 π

x + π - (x - π)

- (x - π) : - x + π

- (x - π)

Distribute parentheses

= - (x) - (- π)

Apply minus-plus rules

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

= - x + π

= x + π - x + π

Simplify

x + π - x + π : 2 π

x + π - x + π

Group like terms

= x - x + π + π

Add similar elements:

x - x = 0

= π + π

Add similar elements:

π + π = 2 π

= 2 π

= 2 π

= \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

= 2 cos (π) sin (x)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

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

= 2 (- 1) sin (x)

Refine

= - 2 sin (x)

= - 2 sin (x)

- 2 sin (x) = 0

Divide both sides by

- 2

- 2 sin (x) = 0

Divide both sides by

- 2

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

Simplify

sin (x) = 0

sin (x) = 0

General solutions for

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Popular Examples

cos(y)-sin(y)=0

cos (y) - sin (y) = 0

36sin(x)cos(x)=-9sqrt(3)

36 sin (x) cos (x) = - 93

sin(2x)=cos(4x)

sin (2 x) = cos (4 x)

6cos^2(θ)+7=-17cos(θ)

6 cos^{2} (θ) + 7 = - 17 cos (θ)

0=-9*sin(2x)

0 = - 9 \cdot sin (2 x)

Frequently Asked Questions (FAQ)

What is the general solution for sin(x+pi)+sin(x-pi)=0 ?
The general solution for sin(x+pi)+sin(x-pi)=0 is x=2pin,x=pi+2pin