What is the general solution for solvefor L,sin(kL)=0 ?

The general solution for solvefor L,sin(kL)=0 is L=(2pin)/k ,L= pi/k+(2pin)/k ;\quad k\ne 0

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solvefor L,sin(kL)=0

solve for

L, sin (k L) = 0

L = \frac{2 πn}{k}, L = \frac{π}{k} + \frac{2 πn}{k}; k \neq = 0

Solution steps

sin (k L) = 0

General solutions for

sin (k L) = 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}

k L = 0 + 2 πn, k L = π + 2 πn

k L = 0 + 2 πn, k L = π + 2 πn

Solve

k L = 0 + 2 πn : L = \frac{2 πn}{k}; k \neq = 0

k L = 0 + 2 πn

0 + 2 πn = 2 πn

k L = 2 πn

Divide both sides by

k; k \neq = 0

k L = 2 πn

Divide both sides by

k; k \neq = 0

\frac{k L}{k} = \frac{2 πn}{k}; k \neq = 0

Simplify

L = \frac{2 πn}{k}; k \neq = 0

L = \frac{2 πn}{k}; k \neq = 0

Solve

k L = π + 2 πn : L = \frac{π}{k} + \frac{2 πn}{k}; k \neq = 0

k L = π + 2 πn

Divide both sides by

k; k \neq = 0

k L = π + 2 πn

Divide both sides by

k; k \neq = 0

\frac{k L}{k} = \frac{π}{k} + \frac{2 πn}{k}; k \neq = 0

Simplify

L = \frac{π}{k} + \frac{2 πn}{k}; k \neq = 0

L = \frac{π}{k} + \frac{2 πn}{k}; k \neq = 0

L = \frac{2 πn}{k}, L = \frac{π}{k} + \frac{2 πn}{k}; k \neq = 0

What is the general solution for solvefor L,sin(kL)=0 ?
The general solution for solvefor L,sin(kL)=0 is L=(2pin)/k ,L= pi/k+(2pin)/k ;\quad k\ne 0