Physic Labs
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
sin⁡(k⋅0)=0 ✓,sin⁡(kL)=0\sin(k \cdot 0) = 0 \ \checkmark, \qquad \sin(kL) = 0
problems.proof.analysis

The condition at x=0x=0 is automatically satisfied since sin⁡(0)=0\sin(0)=0. The condition at x=Lx=L requires sin⁡(kL)=0\sin(kL) = 0 — this is the real constraint; not every kk (every wavelength) satisfies it.

problems.proof.pitfall. Many people forget to check the first condition (x=0x=0) and think both are 'hard', but really only one constraint actually needs solving.