How can I convert a complex number to cartesian form using the unit circle?

[Ry122]

I need some help with converting this to cartesian form.
z=-1+1i
On a graph the relation is (-1,1)
Then I use the pythagorean theorem to find the hypotenuse which works out to be the square root of 2. How do I then find what the angle of the triangle is using the unit circle?

 

[Gib Z]

You already have it in Cartesian form, you want to change it into polar. Yes, you got the magnitude correctly. You also know the length of the sides. Use some simple trig perhaps?

 

[Ry122]

I need to be able to do it without a calculator. All I can use is the values on a unit circle. The problem I am having is that the unit circle shows values for triangles with a hypotenuse of 1 but the value I have here is square root of 2.

 

[Gib Z]

You don't really need a calculator. You actually don't even need trig for this one. Your triangle has 2 sides of the same length, and one angle is 90. What are the other two angles then?

 

[Ry122]

I know they are 45. I just thought there was a way to do it with the unit circle.

 

[Tedjn]

The unit circle is based off of right triangles. In your case, you just need to scale the unit circle by radical 2 since that is your hypotenuse. Everything is increased by a factor radical 2, so the legs of whatever right triangle you drew on the unit circle would be increased by a factor of radical 2 also. But, essentially, it is exactly the same as Gib Z's advice. You should think about it from the point of view of right triangles, and anything to do with the unit circle follows from that.