If I'm asked to find a subscript capital N and a subscript capital T, what does that mean in Calculus? Never seen this before.?

aT ≡ the tangential scalar component of acceleration
aN ≡ the normal scalar component of acceleration

r(t) = < t - sin t , 1 - cos t >

v(t) = dr/dt = < 1 - cos t , sin t >
∣ v ∣ = √[ ( 1 - cos t )^2 + sin^2 t ] = √( 1 - 2*cos t + cos^2 t + sin^2 t ) = √( 2 - 2*cos t )

a(t) = dv/dt = < sin t , cos t >
∣ a ∣ = √( sin^2 t + cos^2 t ) = 1

aT
= d ∣ v ∣ / dt
= (1/2) * ( 2 - 2*cos t )^(-1/2) * ( 2*sin t )
= sin t / √( 2 - 2*cos t )

aT ( π/2 ) = sin ( π/2 ) / √[ 2 - 2*cos ( π/2 ) ] = 1 / √2 = √2 / 2
aT ( 3π/2 ) = - 1 / √2 = - √2 / 2
aT ( π ) = 0

aN = √ ( ∣ a ∣^2 - aT^2 ) = √ ( 1 - aT^2 )
aN ( π/2 ) = √ ( 1 - 1/2 ) = √(1/2) = √2 / 2
aN ( 3π/2 ) = √ ( 1 - 1/2 ) = √(1/2) = √2 / 2
aN ( π ) = √ ( 1 - 0 ) = 1

Ans :
aT ( π/2 ) = √2 / 2 , aT ( 3π/2 ) = - √2 / 2 , aT ( π ) = 0
aN ( π/2 ) = √2 / 2 , aN ( 3π/2 ) = √2 / 2 , aN ( π ) = 1

Fantastic! Thank you so much.