Question Video: Writing a Complex Number in Polar Form given Its Modulus and Principal Argument Mathematics • 12th Grade

Given that |𝑍| = 8 and the argument of 𝑍 is 𝜃 = 360°, find 𝑍, giving your answer in trigonometric form.

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Video Transcript

Given that the modulus of 𝑍 is equal to eight and the argument of 𝑍 is 𝜃 equals 360 degrees, find 𝑍, giving your answer in trigonometric form.

When we write a complex number in trigonometric or polar form, we write it as 𝑍 is equal to 𝑟 multiplied by cos 𝜃 plus 𝑖 sin 𝜃, where 𝑟 is the modulus of the complex number 𝑍 and 𝜃 is its argument. In polar form, 𝜃 can be in degrees or radians, though radians is often preferred, whereas in exponential form, it does need to be in radians.

We’re going to substitute what we know about the complex number 𝑍 into this formula. But before we do, let’s convert the argument into radians. To change from degrees into radians, we can multiply by 𝜋 over 180. 360 multiplied by 𝜋 over 180 is two 𝜋. So 360 degrees is two 𝜋 radians, although this is a standard result we should’ve known by heart.

So we can now substitute what we know into the formula for the trigonometric form or the polar form of a complex number. And doing so, we can see that 𝑍 is equal to eight multiplied by cos of two 𝜋 plus 𝑖 sin of two 𝜋.