# Question Video: Multiplying Complex Numbers in Polar Form Mathematics

If πβ = 7 (cos πβ + π sin πβ), πβ = 16 (cos πβ + π sin πβ), and πβ + πβ = π, then what is πβ πβ?

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### Video Transcript

If π one is equal to seven multiplied by cos π one plus π sin π one, π two is equal to 16 multiplied by cos π two plus π sin π two, and π one plus π two is equal to π, then what is π one multiplied by π two?

Letβs begin by recalling the product formula since weβre going to be finding the product of these two complex numbers. For two complex numbers π one and π two, their product can be found by multiplying their moduli and adding their arguments as shown.

Letβs begin then by applying the product formula to our two complex numbers. The modulus of π one is seven, and the modulus of π two is 16. The product of their two moduli then is found by multiplying seven by 16. The argument of π one is π one, and the argument for π two is π two. This means we can find the argument for π one π two by adding π one and π two.

Seven multiplied by 16 is 112. Weβre also told that π one plus π two is equal to π. So we can rewrite our expression for the product of π one and π two as 112 multiplied by cos π plus π sin π. We know though that cos of π is negative one and sin of π is zero. So we can further rewrite this expression as 112 multiplied by negative one plus zero, which is negative 112. The product of these two complex numbers, π one and π two, is negative 112.