Given complex numbers $z_{1}$ and $z_{2}$,
\[
z_{1}=15\left(\cos 57^{\circ}+i \sin 57^{\circ}\right), \quad z_{2}=38\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)
\]
Give an exact answer. Express any numbers as integers or simplified fractions. Use the degree symbol where necessary.
(a) Find $z_{1} z_{2}$ and write the product in polar form with $0 \leq \theta \leq 360^{\circ}$.
The polar form is

Step 1 ：Given complex numbers $z_{1}$ and $z_{2}$,

Step 2 ：\[z_{1}=15\left(\cos 57^{\circ}+i \sin 57^{\circ}\right), \quad z_{2}=38\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)\]

Step 3 ：We are asked to find $z_{1} z_{2}$ and write the product in polar form with $0 \leq \theta \leq 360^{\circ}$.

Step 4 ：The product of two complex numbers in polar form is given by multiplying their magnitudes and adding their angles.

Step 5 ：So, we need to multiply the magnitudes 15 and 38, and add the angles 57 degrees and 50 degrees.

Step 6 ：\[\text{mag}_\text{z1} = 15, \quad \text{angle}_\text{z1} = 57, \quad \text{mag}_\text{z2} = 38, \quad \text{angle}_\text{z2} = 50\]

Step 7 ：\[\text{mag}_\text{product} = 570, \quad \text{angle}_\text{product} = 107\]

Step 8 ：The magnitude of the product is 570 and the angle is 107 degrees. This is within the range $0 \leq \theta \leq 360^{\circ}$, so no further adjustment is needed.

Step 9 ：\[\boxed{\text{Final Answer: The product } z_{1} z_{2} \text{ in polar form is } 570\left(\cos 107^{\circ}+i \sin 107^{\circ}\right)}\]