What is the general solution for sin((ax)/2)=0 ?

The general solution for sin((ax)/2)=0 is x=(4pin)/a ,x=(2pi)/a+(4pin)/a ;\quad a\ne 0

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sin((ax)/2)=0

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

x = \frac{4 πn}{a}, x = \frac{2 π}{a} + \frac{4 πn}{a}; a \neq = 0

Solution steps

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

General solutions for

sin (\frac{a x}{2}) = 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}

\frac{a x}{2} = 0 + 2 πn, \frac{a x}{2} = π + 2 πn

\frac{a x}{2} = 0 + 2 πn, \frac{a x}{2} = π + 2 πn

Solve

\frac{a x}{2} = 0 + 2 πn : x = \frac{4 πn}{a}; a \neq = 0

\frac{a x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{a x}{2} = 2 πn

Multiply both sides by

2

\frac{a x}{2} = 2 πn

Multiply both sides by

2

\frac{2 a x}{2} = 2 \cdot 2 πn

Simplify

a x = 4 πn

a x = 4 πn

Divide both sides by

a; a \neq = 0

a x = 4 πn

Divide both sides by

a; a \neq = 0

\frac{a x}{a} = \frac{4 πn}{a}; a \neq = 0

Simplify

x = \frac{4 πn}{a}; a \neq = 0

x = \frac{4 πn}{a}; a \neq = 0

Solve

\frac{a x}{2} = π + 2 πn : x = \frac{2 π}{a} + \frac{4 πn}{a}; a \neq = 0

\frac{a x}{2} = π + 2 πn

Multiply both sides by

2

\frac{a x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 a x}{2} = 2 π + 2 \cdot 2 πn

Simplify

a x = 2 π + 4 πn

a x = 2 π + 4 πn

Divide both sides by

a; a \neq = 0

a x = 2 π + 4 πn

Divide both sides by

a; a \neq = 0

\frac{a x}{a} = \frac{2 π}{a} + \frac{4 πn}{a}; a \neq = 0

Simplify

x = \frac{2 π}{a} + \frac{4 πn}{a}; a \neq = 0

x = \frac{2 π}{a} + \frac{4 πn}{a}; a \neq = 0

x = \frac{4 πn}{a}, x = \frac{2 π}{a} + \frac{4 πn}{a}; a \neq = 0

\frac{3}{3} = - \frac{1}{3} \cdot tan (θ)

tan (x + \frac{π}{3}) = 3

16 sin^{2} (x) = 1

x, - 1 \cdot \frac{cos ( x )}{sin ( x )} = 0

8 cos (θ) - 4 = 0

What is the general solution for sin((ax)/2)=0 ?
The general solution for sin((ax)/2)=0 is x=(4pin)/a ,x=(2pi)/a+(4pin)/a ;\quad a\ne 0