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

Given that |๐‘| = 8 and the argument of ๐‘ is ๐œƒ = 360ยฐ, find ๐‘, giving your answer in trigonometric form.

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Video Transcript

Given that the modulus of ๐‘ is equal to eight and the argument of ๐‘ is ๐œƒ equals 360 degrees, find ๐‘, giving your answer in trigonometric form.

When we write a complex number in trigonometric or polar form, we write it as ๐‘ is equal to ๐‘Ÿ multiplied by cos ๐œƒ plus ๐‘– sin ๐œƒ, where ๐‘Ÿ is the modulus of the complex number ๐‘ and ๐œƒ is its argument. In polar form, ๐œƒ can be in degrees or radians, though radians is often preferred, whereas in exponential form, it does need to be in radians.

Weโ€™re going to substitute what we know about the complex number ๐‘ into this formula. But before we do, letโ€™s convert the argument into radians. To change from degrees into radians, we can multiply by ๐œ‹ over 180. 360 multiplied by ๐œ‹ over 180 is two ๐œ‹. So 360 degrees is two ๐œ‹ radians, although this is a standard result we shouldโ€™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 ๐‘ is equal to eight multiplied by cos of two ๐œ‹ plus ๐‘– sin of two ๐œ‹.

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