"Can you find the product of z1 and..." solved and explained

Zachariah Mcmillan

Answered question

2022-12-28

Can you find the product of

z_{1}

and

z_{2}

z_{1} = 4 (\cos 40 ˚ + i \sin 40 ˚)

z_{2} = 2 (\cos 20 ˚ + i \sin 20 ˚)

Give your answer in rectangular form.

Bellitifya

Beginner2022-12-29Added 6 answers

There are two methods to accomplish this.

z_{1} = r_{1} (\cos θ + i \sin θ)

z_{2} = r_{2} (\cos ϕ + i \sin ϕ)

\Rightarrow z_{1} z_{2} = r_{1} r_{2} (\cos (θ + ϕ) + i \sin (θ + ϕ)) - - (1)

so we have:

z_{1} = 4 (\cos 40 + i \sin 40)

z_{2} = 2 (\cos 20 + i \sin 20)

∴ z_{1} z_{2} = (4 \times 2) (\cos (40 + 20) + i \sin (40 + 20))

z_{1} z_{2} = 8 (\cos 60 + i \sin 60)

z_{1} z_{2} = 8 (\frac{1}{2} + i \frac{\sqrt{3}}{2})

z_{1} z_{2} = 4 + 4 \sqrt{3} i

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