Converting a complex number from polar to rectangular form | Precalculus | Khan Academy

We are told to consider the complex number ( z ), which is equal to the square root of 17 times cosine of 346 degrees plus ( i ) sine of 346 degrees. They ask us to plot ( z ) in the complex plane below. If necessary, round the point coordinates to the nearest integer.

So I encourage you to pause this video and at least think about where we would likely plot this complex number.

All right, now let's work through it together. When you look at it like this, you can see that what's being attempted is a conversion from polar form to rectangular form. If we're thinking about polar form, we can think about the angle of this complex number, which is clearly 346 degrees.

346 degrees would be about... would be about 14 degrees short of a full circle, so it would get us probably something around there. We also see what the magnitude or the modulus of the complex number is right over here: square root of 17.

Square root of 17 is a little bit more than 4 because 4 squared is 16. So if we go in this direction, let's see... that's going to be about 1, 2, 3, 4. We're going to go right about there.

So if I were to just guess where this is going to put us, it's going to put us right around here—right around ( 4 - i ). But let's actually get a calculator out and see if this evaluates to roughly ( 4 - i ).

So for the real part, let's go 346 degrees, and we're going to take the cosine of it, and then we're going to multiply that times the square root of 17. So times 17 square root... a little over four, which is equal to that; actually, yes, the real part does look almost exactly four, especially if we are rounding to the nearest integer; it's a little bit more than four.

Now let's do the imaginary part. So we have 346 degrees, and we're going to take the sine of it, and we're going to multiply that times the square root of 17 times 17 square root... which is equal to... yup, if we were to round to the nearest integer, it's about negative 1.

So we get to this point right over here, which is approximately ( 4 - i ), and we are done.