What is the general solution for sin(pix)=-sin(pi-pix) ?

The general solution for sin(pix)=-sin(pi-pix) is x=2n,x=1+2n

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sin(pix)=-sin(pi-pix)

Solution

sin (π x) = - sin (π - π x)

Solution

x = 2 n, x = 1 + 2 n

Degrees

x = 0^{\circ} + 114.59155 \dots^{\circ} n, x = 57.29577 \dots^{\circ} + 114.59155 \dots^{\circ} n

Solution steps

sin (π x) = - sin (π - π x)

Rewrite using trig identities

sin (π x) = - sin (π - π x)

Rewrite using trig identities

sin (π - π x)

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (π) cos (π x) - cos (π) sin (π x)

Simplify

sin (π) cos (π x) - cos (π) sin (π x) : sin (π x)

sin (π) cos (π x) - cos (π) sin (π x)

sin (π) cos (π x) = 0

sin (π) cos (π x)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 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}

= 0

= 0 \cdot cos (π x)

Apply rule

0 \cdot a = 0

= 0

cos (π) sin (π x) = - sin (π x)

cos (π) sin (π 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

= - 1 \cdot sin (π x)

Multiply:

1 \cdot sin (π x) = sin (π x)

= - sin (π x)

= 0 - (- sin (π x))

Refine

= sin (π x)

= sin (π x)

sin (π x) = - sin (π x)

sin (π x) = - sin (π x)

Subtract

- sin (π x)

from both sides

2 sin (π x) = 0

Divide both sides by

2

2 sin (π x) = 0

Divide both sides by

2

\frac{2 sin ( π x )}{2} = \frac{0}{2}

Simplify

sin (π x) = 0

sin (π x) = 0

General solutions for

sin (π 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}

π x = 0 + 2 πn, π x = π + 2 πn

π x = 0 + 2 πn, π x = π + 2 πn

Solve

π x = 0 + 2 πn : x = 2 n

π x = 0 + 2 πn

0 + 2 πn = 2 πn

π x = 2 πn

Divide both sides by

π

π x = 2 πn

Divide both sides by

π

\frac{π x}{π} = \frac{2 πn}{π}

Simplify

x = 2 n

x = 2 n

Solve

π x = π + 2 πn : x = 1 + 2 n

π x = π + 2 πn

Divide both sides by

π

π x = π + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{π}{π} + \frac{2 πn}{π} : 1 + 2 n

\frac{π}{π} + \frac{2 πn}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{2 πn}{π}

Cancel

\frac{2 πn}{π} : 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= 1 + 2 n

x = 1 + 2 n

x = 1 + 2 n

x = 1 + 2 n

x = 2 n, x = 1 + 2 n

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Frequently Asked Questions (FAQ)

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