Given complex numbers $z_{1}$ and $z_{2}$,
\[
z_{1}=1\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right), z_{2}=4\left(\cos \frac{7 \pi}{12}+i \sin \frac{7 \pi}{12}\right)
\]

Give an exact answer. Express any numbers as integers or simplified fractions.
Part 1 of 2
(a) Find $z_{1} z_{2}$ and write the product in polar form with $0 \leq \theta \leq 2 \pi$.

Step 1 ：Given complex numbers \(z_{1}\) and \(z_{2}\), where \(z_{1}=1\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)\) and \(z_{2}=4\left(\cos \frac{7 \pi}{12}+i \sin \frac{7 \pi}{12}\right)\)

Step 2 ：We are asked to find the product \(z_{1} z_{2}\) and write it in polar form with \(0 \leq \theta \leq 2 \pi\)

Step 3 ：The product of two complex numbers in polar form is given by the product of their magnitudes and the sum of their angles

Step 4 ：The magnitude of \(z_{1} z_{2}\) is \(1*4=4\)

Step 5 ：The angle is \(\frac{7 \pi}{4} + \frac{7 \pi}{12}\)

Step 6 ：We need to simplify the angle to fall within the range \(0 \leq \theta \leq 2 \pi\)

Step 7 ：The simplified angle is \(\frac{\pi}{3}\)

Step 8 ：\(\boxed{The product \(z_{1} z_{2}\) in polar form is \(4\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\)}