Converting Complex Numbers From Trigonometric Form to Rectangular - Problem 2

I'm converting numbers from trigonometric form to rectangular form. Here I have a complex number 5 root 2, times the quantity cosine 7 pi over 4, plus i sine 7 pi over 4.

Just to refresh you memory, what does this mean? If I were to graph this number, the 5 root 2 is the distance of the number from the origin. The 7 pi over 4 is the argument of the number, which is its angle from the positive real axis, this axis. So 7 pi over 4 is all the way over here, leaving a remainder of pi over 4.

So this is going to be our point 7 pi over 4, and this length is 5 root 2. That's our complex number. Now let's convert it to rectangular form. I'll distribute the 5 root 2. I get 5 root 2 cosine 7 pi over 4, plus i times 5 root 2, sine 7 pi over 4. Now cosine 7 pi over 4 is root 2 over 2. So this is 5 root 2, times root 2 over 2 plus, now sine of 7 pi over 4 is negative root 2 over 2. So 5 root 2 times negative root 2 over 2. Now here root 2 times root 2 is 2, divided by 2 is 1. So this whole thing is just going to be 5.

Then over here, the same thing. This is really the same product only there's a minus sign. So I'll get minus 5i. Z equals 5 minus 5i. So those would be the coordinates of this point, 5 minus 5i. That means it's 5 units to the right, and 5 down. That's the rectangular form of z.