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

In summary, the conversation is about converting a complex number into Cartesian form and finding the angle using the unit circle. The person needs to find the angle without using a calculator and is struggling because the unit circle is based on right triangles with a hypotenuse of 1. They are advised to scale the unit circle by the square root of 2, which is the hypotenuse in this case, and everything else will also increase by the same factor. Essentially, it is the same concept as using right triangles to find the angle on the unit circle.
  • #1
Ry122
565
2
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?
 
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  • #2
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?
 
  • #3
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.
 
  • #4
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?
 
  • #5
I know they are 45. I just thought there was a way to do it with the unit circle.
 
  • #6
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.
 

1. What are complex numbers?

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (defined as the square root of -1).

2. What is the meaning of the term "z=-1+1i"?

The term "z=-1+1i" is a complex number in the form a + bi, where a = -1 and b = 1. This means that the number has a real part of -1 and an imaginary part of 1i.

3. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply add or subtract the real parts and the imaginary parts separately. For example, to add z1 = a1 + b1i and z2 = a2 + b2i, you would add a1 and a2 to get the real part, and add b1i and b2i to get the imaginary part.

4. Can complex numbers be graphed on a traditional x-y coordinate plane?

Yes, complex numbers can be graphed on a traditional x-y coordinate plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The point (-1,1) on the complex plane would represent the complex number z=-1+1i.

5. What are the practical applications of complex numbers?

Complex numbers have many practical applications in mathematics, engineering, physics, and other sciences. They are used to describe electrical circuits, vibrations, and other physical phenomena. They are also used in signal processing, control systems, and other areas of science and technology.

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