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

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

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sin(ax)=0

sin (a x) = 0

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

Solution steps

sin (a x) = 0

General solutions for

sin (a 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}

a x = 0 + 2 πn, a x = π + 2 πn

a x = 0 + 2 πn, a x = π + 2 πn

Solve

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

a x = 0 + 2 πn

0 + 2 πn = 2 πn

a x = 2 πn

Divide both sides by

a; a \neq = 0

a x = 2 πn

Divide both sides by

a; a \neq = 0

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

Simplify

x = \frac{2 πn}{a}; a \neq = 0

x = \frac{2 πn}{a}; a \neq = 0

Solve

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

a x = π + 2 πn

Divide both sides by

a; a \neq = 0

a x = π + 2 πn

Divide both sides by

a; a \neq = 0

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

Simplify

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

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

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

cos (x + 3 0^{\circ}) = 2 cos (x)

x, sec (x) tan (x) = 23

4 sin (θ) - 4 cos (θ) = 2

4 sin (x) + 22 = 0

sin (4 x) + sin (2 x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

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