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

The general solution for sin(wpi)=0 is w=2n,w=1+2n

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sin(wpi)=0

sin (w π) = 0

w = 2 n, w = 1 + 2 n

Degrees

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

Solution steps

sin (w π) = 0

General solutions for

sin (w 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}

w π = 0 + 2 πn, w π = π + 2 πn

w π = 0 + 2 πn, w π = π + 2 πn

Solve

w π = 0 + 2 πn : w = 2 n

w π = 0 + 2 πn

0 + 2 πn = 2 πn

w π = 2 πn

Divide both sides by

π

w π = 2 πn

Divide both sides by

π

\frac{w π}{π} = \frac{2 πn}{π}

Simplify

w = 2 n

w = 2 n

Solve

w π = π + 2 πn : w = 1 + 2 n

w π = π + 2 πn

Divide both sides by

π

w π = π + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{w π}{π} : w

\frac{w π}{π}

Cancel the common factor:

π

= w

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

w = 1 + 2 n

w = 1 + 2 n

w = 1 + 2 n

w = 2 n, w = 1 + 2 n

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

cos (x) = - 8

arctan (\frac{18.5}{x}) - arctan (\frac{42.41}{x}) = 21.27

2 sin (2 x) = sin^{2} (x)

5 sin^{2} (x) = 3 sin (x) + 2

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