Polar Coordinates and Complex Numbers
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 we must understand how to convert from rectangular form to trigonometric form. To convert from rectangular form to trigonometric form we need to calculate the modulus and the angle of the position vector. It is also important to be able to convert from trigonometric to rectangular form.
We're converting from rectangular form to trigonometric form and we're starting with the complex number z equals negative root 2 plus i times root 2. Now, first of all, what does it mean to convert to trigonometric form?
Well, I have my number in rectangular form, so it's in a+bi form. This is the a value, this is the b value. The a's the real part of the imaginary number and root 2 is the the imaginary part. Now if I were to plot this, you'd have a is negative and b is positive so you'd be somewhere in the second quadrant. And I want to choose theta, right? When i get into trigonometric form, theta is the argument of the complex number. I want to choose theta so that it's between 0 and 2 pi. So I'd want to choose this theta and r is the distance between the point and the origin. That's the absolute value of the number. r is the easiest to get because we have this formula r equals the square root of a squared plus b squared, so let's get r first.
The square root of negative root 2 squared plus positive root 2 squared. So this is going to end up being 2 plus 2, 4. Root 4 which is 2, so r=2.
Next, let's try to find theta. That's the harder thing to find but we can use these 2 conversions to find theta. I'm going to use the fact that cosine theta would equal a over r. So cosine theta equals a over r. a is negative root 2 and r is 2. And we also use the fact that sine theta is b over r. Sine theta is b over r. b is root 2. r is 2.
So we need to find, figure out what theta is going to have a cosine of negative root 2 over 2 and a sine of root 2 over 2. Now the numbers root 2 over 2, you may remember are the sine and cosine of pi over 4, pi over 4's our reference angle. That means this angle is pi over 4 and that means this angle is going to be 3 pi over 4. So theta is 3 pi over 4. That's our argument of z. And so we put it into trig form. r times the quantity cosine of theta plus i sine theta. So z=r which is 2 cosine of 3 pi over 4 plus i times the sine of 3 pi over 4.
Don't forget when you're in trig form this number will always be the same for both cosine and sine, right? This is the argument of the complex number. This is our final answer for the trig form of the number.