A thin plate in a machine that sorts small parts vibrates along a straight line, and a camera…

A thin plate in a machine that sorts small parts vibrates along a straight line, and a camera records its position at several times. The velocity of the plate at time $t$ seconds is $v(t) = \sin 4t\cos 2t$ centimetres per second, and at $t = 0$ the plate is at its starting point, so the displacement $y$ satisfies $y(0) = 0\ \text{cm}$. The engineer sees a product of a sine and a cosine with different multiples of $t$. She uses the product formula $2\sin A\cos B = \sin(A + B) + \sin(A - B)$ to turn the velocity into a sum, and then integrates term by term. She writes the displacement as $y(t) = -\dfrac{\cos 6t}{12} - \dfrac{\cos 2t}{4} + C$. Find the constant $C$ in centimetres. Then find the displacement at $t = \dfrac{\pi}{6}\ \text{s}$ and at $t = \dfrac{\pi}{2}\ \text{s}$. Give each answer in centimetres, to four decimal places.