Question Video: Multiplying Complex Numbers in Polar Form | Nagwa Question Video: Multiplying Complex Numbers in Polar Form | Nagwa

# Question Video: Multiplying Complex Numbers in Polar Form Mathematics • Third Year of Secondary School

## Join Nagwa Classes

Given that π§β = 6 (cos 4π + π sin 4π) and π§β = 1/3 (sin 2π + π cos 2π), where 0 < π < 90Β°, determine the trigonometric form of π§βπ§β.

02:14

### Video Transcript

Given that π§ one is equal to six cos four π plus π sin four π and π§ two is equal to a third of sin π plus π cos two π, where zero is less than π which is less than 90 degrees, determine the trigonometric form of π§ one π§ two.

A complex number, is in polar form. If it looks like this π§ is equal to π cos π plus π sin π. Notice that our second complex number is not in this form. So weβll first need to perform some clever manipulation to transform it. Recall the relationship between the sine and cosine curve. Theyβre translations of one another such that sin of π is equal to cos of 90 minus π.

We can therefore say that sin of two π must be equal to cos of 90 minus two π. We also know that cos of π is equal to sin of 90 minus π. So that means that cos of two π must be equal to sin of 90 minus two π. And we can therefore write π§ two now as a third of cos of 90 minus two π plus π sin of 90 minus two π. We need to find the product of π§ one and π§ two. Recall the product formula. This says that for two complex numbers expressed in polar form, π§ one with a modulus of π one and an argument of π one and π§ two with a modulus of π two and an argument of π two, their product can be found by multiplying their moduli and adding their arguments.

Thatβs π one π two multiplied by cos of π one plus π two plus π sin of π one plus π two. When we multiply the moduli of our two complex numbers together. Thatβs six multiplied by one third, which is two. Adding their arguments we get four π plus 90 minus two π, which is equal to 90 plus two π. And we can therefore say the trigonometrical polar form of π§ one π§ two is two multiplied by cos of 90 degrees plus two π plus π sin of 90 degrees plus two π.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions