Video: Multiplying Complex Numbers in Polar Form

Simplify 4 (cos 90ยฐ + ๐‘– sin 90ยฐ) ร— 5 (cos 80ยฐ + ๐‘– sin 80ยฐ) ร— 4 (cos 45ยฐ + ๐‘– sin 45ยฐ), giving your answer in trigonometric form.


Video Transcript

Simplify four cos of 90 plus ๐‘– sin of 90 multiplied by five cos of 80 plus ๐‘– sin 80 multiplied by four cos of 45 plus ๐‘– sin of 45, giving your answer in trigonometric form.

Recall the product formula. This says that, for two complex numbers expressed in polar form, ๐‘ one with a modulus of ๐‘Ÿ one and an argument of ๐œƒ one and ๐‘ two with a modulus of ๐‘Ÿ two and an argument of ๐œƒ two, their product can be found by multiplying the moduli and adding the arguments.

We can extend this to three complex numbers and find the product of the three complex numbers that weโ€™ve been given. Letโ€™s begin by multiplying their moduli. Thatโ€™s four, five, and four, which is 80. Next, weโ€™ll add their arguments. Thatโ€™s 90, 80, and 45, which is 215. In trigonometric or polar form then, the product of these three complex numbers is 80 multiplied by cos of 215 degrees plus ๐‘– sin of 215.

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