Taylor polynomials practice question 12

Here $\theta$ is $T_{2}$, not only $T_{1}$. Which derivative, which factorial and which power of the step does the remainder after $T_{2}$ use?

With $f(\theta) = \sin\theta$, the remainder after $T_{2}$ is $\dfrac{f'''(c)}{3!}\,\theta^{3}$ for some $c$ between 0 and $\theta$.

$f'''(\theta) = -\cos\theta$. How large can $|\cos c|$ be for $c$ in $(0, 0.2)$?