Question

State true or false
If sin x = sin$\lambda$, then the values of sin(x/3) are sin ($\lambda$/3), sin [$(\pi -\lambda )$ /3] and - sin [$(\pi +\lambda )$ /3]

• A. True
• B. False

Answer 1

The correct option is A True

We have $\sin x-\sin \lambda$
$\Rightarrow x=n\pi +(-1{)}^n\lambda ,nϵl$
$\therefore$ $\sin \left(x/3\right)=\sin [n\pi /3+(-1{)}^2\left(X/3\right)]$
Hence for n = 0,3, we have
$\sin \left(x/3\right)=\sin \left(\lambda /3\right)$
and for n = 1,2, we have
$\sin \left(x/3\right)$ = $\sin \pi /3-\lambda /3$
[Note that for n = 2, sin(x /3)
=$\sin (2\pi /3+\lambda /3)$ = $\sin [\pi -[\pi /3-\lambda /3\left]\right]$
= $\sin (\pi /3-\lambda /3)$
and for n = 4,5, we get
sin (x/3) = -$\sin (\pi /3+\lambda /3)$.
It is easy to see that all other values of n repeat only these values of $\sin \left(x/3\right)$.