A particle executes simple harmonic motion and it is located at x =...

$a = Asin \omega{t}_{0}$...$( 1 )$
$b = Asin 2\omega{t}_{0}$...$( 2 )$
$c = Asin 3\omega{t}_{0}$...$( 3 )$

adding equation $( 1 )$ and $( 3 )$

$a + c = A\left[sin \omega{t}_{0} + \sin 3\omega{t}_{0}\right] = 2Asin 2\omega{t}_{0}cos \omega{t}_{0}$...$( 4 )$

$∵ sinC + sinD = 2sin\dfrac{C + D}{2}cos\dfrac{C - D}{2}$

from equation $( 2 )$ and $( 4 )$

$\dfrac{a + c}{b} = 2cos\omega{t}_{0}$

$\omega = \dfrac{1}{{t}_{0}}\cos^{- 1} \left(\dfrac{a + c}{2b}\right) ⇒ f = \dfrac{1}{2\pi{t}_{0}}\cos^{- 1} \left(\dfrac{a + c}{2b}\right)$