Let sin 22° = p. Determine the following ratios in terms of p: 2.2.1 sin 202°, 2.2.2 cos 44°

The angle $202^\circ$ lies in the third quadrant. The reference angle is $202^\circ - 180^\circ = 22^\circ$.
In the third quadrant, the sine function is negative.

$$\sin 202^\circ = \sin (180^\circ + 22^\circ) = -\sin 22^\circ$$

Since $\sin 22^\circ = p$, we have:

$$\sin 202^\circ = -p$$

The final answer is $\boxed{\boldsymbol{-p}}$.

2.2.2

We know that $44^\circ = 2 \times 22^\circ$. We can use the double angle identity for cosine, $\cos 2\theta = 1 - 2\sin^2 \theta$.
Let $\theta = 22^\circ$.

$$\cos 44^\circ = \cos (2 \times 22^\circ) = 1 - 2\sin^2 22^\circ$$

Since $\sin 22^\circ = p$, we have:

$$\cos 44^\circ = 1 - 2(p)^2 = 1 - 2p^2$$

The final answer is $\boxed{\boldsymbol{1 - 2p^2}}$.