A Cepheid variable star is a star whose brightness alternately increases and decreases. Suppose that Cephei Joe is a star for which the interval between times of maximum brightness is $5\mathrm{.}9$ days. Its average brightness is $4\mathrm{.}8$ and the brightness changes by $\frac{+}{-}0\mathrm{.}2.$ Using this data, we can construct a mathematical model for the brightness of Cephei Joe at time $t,$ where $t$ is measured in days:

$B(t)=4\mathrm{.}8+0\mathrm{.}2sin(2\frac{t}{5\mathrm{.}9})$

$($a$)$ Find the rate of change of the brightness after $t$ days.

Rate of change $=$

$($b$)$ Find the rate of increase after one day.

Rate of increase $=$