You said pi/2 at first, then changed to pi/3
I assume that we are talking about a spring here so
PE = (1/2)kx^2
v = 10 pi cos (pi t + pi/2)
v^2 = 100 pi^2 cos^2 (pi t + pi/2)
KE = (1/2) m v^2 = 50 m pi^2 cos^2(pi t +pi/2)
so sort of agree with you
Now PE = (1/2) k x^2 =
(1/2)k 100 sin^2( pi t + pi/2)
so when does PE = KE?
50 m pi^2 cos^2 pi t + pi/2) =50 k sin^2(pi t + pi/2)
m pi^2 cos^2 = k sin^2
tan^2 (pi t + pi/2 ) =pi^2 m/k
tan (pi t +pi/2) = +/- pi sqrt(m/k)
t =-1/2 + (1/pi)tan^-1(pi sqrt(m/k))
t =-1/2 - (1/pi)tan^-1(pi sqrt(m/k))
Now sqrt (m/k) means something
wo the natural frequency = sqrt (k/m)
= 2 pi fo = 2 pi/T where T is the period of free vibration
sqrt (m/k) = T/ 2 pi
so use that
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