Video: Multiplying Complex Numbers in Polar Form

If 𝑍₁ = 7 (cos πœƒβ‚ + 𝑖 sin πœƒβ‚), 𝑍₂ = 16 (cos πœƒβ‚‚ + 𝑖 sin πœƒβ‚‚), and πœƒβ‚ + πœƒβ‚‚ = πœ‹, then what is 𝑍₁ 𝑍₂?

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Video Transcript

If 𝑍 one is equal to seven multiplied by cos πœƒ one plus 𝑖 sin πœƒ one, 𝑍 two is equal to 16 multiplied by cos πœƒ two plus 𝑖 sin πœƒ two, and πœƒ one plus πœƒ two is equal to πœ‹, then what is 𝑍 one multiplied by 𝑍 two?

Let’s begin by recalling the product formula since we’re going to be finding the product of these two complex numbers. For two complex numbers 𝑍 one and 𝑍 two, their product can be found by multiplying their moduli and adding their arguments as shown.

Let’s begin then by applying the product formula to our two complex numbers. The modulus of 𝑍 one is seven, and the modulus of 𝑍 two is 16. The product of their two moduli then is found by multiplying seven by 16. The argument of 𝑍 one is πœƒ one, and the argument for 𝑍 two is πœƒ two. This means we can find the argument for 𝑍 one 𝑍 two by adding πœƒ one and πœƒ two.

Seven multiplied by 16 is 112. We’re also told that πœƒ one plus πœƒ two is equal to πœ‹. So we can rewrite our expression for the product of 𝑍 one and 𝑍 two as 112 multiplied by cos πœ‹ plus 𝑖 sin πœ‹. We know though that cos of πœ‹ is negative one and sin of πœ‹ 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, 𝑍 one and 𝑍 two, is negative 112.

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