Understanding the Unit Circle Group

In summary, the conversation discusses a problem from a book involving the multiplication group U and its subgroup U z_0, with a question about computing U mod U z_0. The first part of the problem is considered trivial, while the second part leads to a discussion about the isomorphism of U/<-1>. The final conclusion is that U/<-1> is isomorphic to U.
  • #1
ehrenfest
2,020
1
[SOLVED] unit circle

Homework Statement


My book contains the following problem:

Let U be the multiplication group [itex] \{z \in C : |z| = 1\} [/itex]

1) Let z_0 be in U. Show that [itex] U z_0 = \{ z z_0 : z \in U \}[/itex] is a subgroup of U, and compute U mod U z_0.
2) To what group is U/<-1> isomorphic to?

Homework Equations


The Attempt at a Solution


I think 1) is so insanely trivial it is not worth asking. The answer is clearly the trivial group, right?

My book says that the answer to 2) is U, but it seems it should be the half-circle or the reals mod 2 or something. Why is it U?[tex]\in\in[/tex]
 
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  • #2
U can also be represented as an additive group. It's R/(Z*2pi). U/<-1> is R/(Z*pi). Is there any real difference? The 'half circle' is isomorphic to the 'full circle'.
 

Related to Understanding the Unit Circle Group

What is the unit circle group?

The unit circle group is a mathematical concept that represents all the possible points that lie on a circle with a radius of 1 unit. It is a group because it has a defined set of operations (such as addition and multiplication) that follow certain rules.

Why is it important to understand the unit circle group?

The unit circle group is important in many areas of mathematics, including trigonometry and complex numbers. It helps us visualize and understand the relationships between angles and coordinates on the circle, and is a fundamental concept in solving equations and proving theorems.

How do you use the unit circle group in trigonometry?

In trigonometry, the unit circle group is used to find the values of trigonometric functions (such as sine, cosine, and tangent) for any angle. By placing an angle on the unit circle, we can determine the x and y coordinates of the corresponding point, which can then be used to calculate the trigonometric function values.

What is the relationship between the unit circle and complex numbers?

The unit circle group and complex numbers are closely related. The coordinates of a point on the unit circle can be represented as a complex number, where the real part is the x coordinate and the imaginary part is the y coordinate. This connection helps us understand the properties of complex numbers and their relationships to trigonometric functions.

How can I remember the values of trigonometric functions on the unit circle?

There are many techniques for remembering the values of trigonometric functions on the unit circle, such as using the acronym SOHCAHTOA (sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent) or creating a mnemonic device. It is also important to practice and understand the patterns and relationships between the values on the unit circle.

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