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# Converting Complex Numbers From Trigonometric Form to Rectangular - Concept 14,514 views

###### Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Both the trigonometric form and the rectangular form are useful ways to describe complex numbers, and so it is important to understand converting complex numbers from trigonometric form to rectangular. The method for converting complex numbers from trigonometric form to rectangular is the reverse of converting from rectangular to trigonometric form.

So we just learned about 2 forms for complex numbers. there's rectangular form, the familiar a+bi form and then there's trigonometric form, z=r times the quantity cosine theta plus i sine theta, where theta is the angle that this line makes with the real axis. This is sometimes called the argument of z and r is the distance of z from the origin and that's called the modulus or absolute value.
Now I've got a number in trig form. Let's convert this to rectangular form. And it's pretty easy, all you have to do is distribute the 6 through and you get 6 cosine pi over 3 plus i times 6 sine pi over 3. Now cosine pi over 3 is a half. So 6 times a half 3. Sine of pi over 3 is root 3 over 2. Root 3 over 2 times 6 is 3 root 3 so it's i times 3 root 3.
Sometimes when there's a radical, teachers like to put the i in front so there's no confusion about whether the i is inside the radical. So this is our final answer. This is the rectangular form of z.