STATEMENT:

It states, "If three forces acting at a point on a body keep it at rest, then each force is proportional to the sine of the angle between the other two forces.

Mathematically,

$\dfrac{P}{\sin\alpha}=\dfrac{Q}{\sin\beta}=\dfrac{R}{\sin\gamma}$

Where,

$\alpha$: Angle between $Q$ and $R$ $\beta $: Angle between $P$ and $R$ $\gamma$: Angle between $P$ and $Q$ $P, Q$ and $R$: Three concurrent forces. To prove: $\dfrac{P}{\sin\alpha}=\dfrac{Q}{\sin\beta}=\dfrac{R}{\sin\gamma}$ Proof:  Construct a parallelogram OACB such that, $OA=P$ $OB=Q$ $OC=R$ In ΔOAC, applying sine rule, we get $\dfrac{OA}{\sin(\pi-\alpha)}=\dfrac{AC}{\sin(\pi-\beta)}=\dfrac{OC}{\sin(\pi-\gamma)}$ $\dfrac{P}{\sin\alpha}=\dfrac{Q}{\sin\beta}=\dfrac{R}{\sin\gamma}=k$ Where $k$ is a constant Therefore, $P=k\sin\alpha$ $Q=k\sin\beta$ $R=k\sin\gamma$

Hence, each force is proportional to the sine of the angle between the two other forces.