What is the general solution for 0=sin(pix) ?

The general solution for 0=sin(pix) is x=2n,x=1+2n

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0=sin(pix)

0 = sin (π x)

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

0 = sin (π x)

Switch sides

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

sec (θ) csc (θ) = 1

cos (a + 27 0^{\circ}) = sin (a)

1 + sin (θ) = 2 - sin (θ)

- 4 cos^{2} (θ) + cos (θ) = 9 cos (θ) + 3

3 + 4 cot (x) = 5 csc (x)

What is the general solution for 0=sin(pix) ?
The general solution for 0=sin(pix) is x=2n,x=1+2n