Given that 𝑧 one is equal to five
multiplied by cos two 𝑎 plus 𝑖 sin two 𝑎 and 𝑧 two is equal to a quarter
multiplied by cos four 𝑎 plus 𝑖 sin four 𝑎, find 𝑧 one 𝑧 two.
Recall the product formula. This says that, for two complex
numbers expressed in polar form, 𝑧 one has a modulus of 𝑟 one and an argument of
𝜃 one and 𝑧 two which has a modulus of 𝑟 two and an argument of 𝜃 two, their
products can be found by multiplying together their moduli and adding together their
arguments. It’s 𝑟 one 𝑟 two multiplied by
cos of 𝜃 one plus 𝜃 two plus 𝑖 sin of 𝜃 one plus 𝜃 two.
In our question, the modulus of 𝑧
one is five and the modulus of 𝑧 two is one-quarter. This means we can find the modulus
of 𝑧 one 𝑧 two by multiplying five by one-quarter, which is five-quarters. The argument of 𝑧 one is two 𝑎,
and the argument of 𝑧 two is four 𝑎. So, we find the sum of their
arguments. That’s two 𝑎 plus four 𝑎, which
is six 𝑎. 𝑧 one 𝑧 two is therefore given as
five-quarters multiplied by cos six 𝑎 plus 𝑖 sin six 𝑎.