# What is Unit circle: Definition and 106 Discussions

In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation

x

2

+

y

2

=
1.

{\displaystyle x^{2}+y^{2}=1.}
Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.
The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.
One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.

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1. ### One-parameter parametrization of a unit circle in R^n

I tried to looking at lower-dimensional cases: For ##n=2## we have $$(x(t),y(t))=(cos(t),sin(t))$$ For ##n=3## we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to $$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$ It seems...
2. ### I Equation to graph a sine wave that acts like a point on a unit circle

I need an equation to graph a sine wave that act like a unit circle but only positive numbers. so I need it to be 0 at 0, A at 90 , 0 at 180, A at 270, 0 at 360, and A at 450 and so on and so on... Now I know sin(0) is 0 in degrees and sin(90) 1 and I know if you Square a number is...
3. ### Digital Filters: why is sampling frequency equal to 2*pi unit circle

Hi, I was working through a filter design problem and got stuck on a concept. Scenario: Let us say we have the following pulse transfer function and the sampling frequency is ## f_s = 50 \text{Hz} ##. G(z) = \frac{1}{3} \left( 1 + z^{-1} + z^{-2} \right) The zeros of the transfer function...
4. ### MHB Draw Angles & Find Values in Unit Circle

Hey! :giggle: Make a drawing for each of the values of the angle below indicating the angle at the unit circle (in other words: $\text{exp} (i \phi )$) and its sine, show cosine, tangent and cotangent. Give these four values explicitly in every case (you are allowed to use elementary...
5. ### B Why does the radius of a unit circle need to be 1?

Why is it that the radius of the unit circle is 1?
6. ### I How to know if a complex root is inside the unit circle

Hi. I have been trying to calculate the real definite integral with limits 2π and 0 of ## 1/(k+sin2θ) ## To avoid the denominator becoming zero I know this means |k|> 1 Making the substitution ##z= e^{iθ}## eventually ends up giving me a quadratic equation in ##z^2## with 2 pairs of roots...
7. ### Complex polynomial on the unit circle

So, the values of polynomial ##p## on the complex unit circle can be written as ##\displaystyle p(e^{i\theta}) = a_0 + a_1 e^{i\theta} + a_2 e^{2i\theta} + \dots + a_n e^{ni\theta}##. (*) If I also write ##\displaystyle a_k = |a_k |e^{i\theta_k}##, then the complex phases of the RHS terms of...
8. ### MHB Graphs of Functions Covering Unit Circle .... Hubbard & Hubbard, Example 3.1.5 .... ....

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Section 3.1: Manifolds ... I need some help in order to understand Example 3.1.3 ... ... Example 3.1.3 reads as follows:In...
9. ### MHB Unit Circle Problems solve cosW=sin20, sinW=cos(-10), sinW< 0.5 and 1<tanW

Without a calculator, find all solutions w between 0 and 360, inclusive, providing diagrams that support your results. 1) cosW=sin20 2) sinW=cos(-10) 3) sinW< 0.5 4) 1<tanW
10. A

### Finding the direction of an angle in the unit circle

Homework Statement I'm having trouble understanding how to find the angle of a vector. Here we are given the x and y component to help us find the direction of vector C. In this case, both x and y component is negative, so it should be in the third quadrant. I know that since we have both the x...
11. ### MHB What is the Sum of Lengths for a Regular n-gon Inscribed in a Unit Circle?

Let $S_n$ be the sum of lengths of all the sides and all the diagonals of a regular $n$-gon inscribed in a unit circle. (a). Find $S_n$. (b). Find $$\lim_{{n}\to{\infty}}\frac{S_n}{n^2}$$
12. ### I Why is this line not homeomorphic to the unit circle?

I've been told that ##[0, 2 \pi )## is not homeomorphic to the unit circle in ##\mathbb{R}^2##. Why not? From intuition, it would seem that I could just bend the line segment to fit the shape of a circle.
13. ### B How Does the Unit Circle Relate to Euler's Formula in Complex Numbers?

Hi everyone. I was looking at complex numbers, eulers formula and the unit circle in the complex plane. Unfortunately I can't figure out what the unit circle is used for. As far as I have understood: All complex numbers with an absolut value of 1 are lying on the circle. But what about...
14. ### I Arc Length Parameterization for Unit Circle: Cos(s) & Sin(s)

(cos(s), sin(s)) gives an arc-length parameterization of the unit circle so that the speed is constantly 1, but the second derivative doesn't give zero acceleration which should be the case with constant speed?
15. ### Finding the Value of axb on the Unit Circle | Round to the Nearest Thousandths

Homework Statement the point (log a, log b) exists on the unit circle. find the value of axb. round to the nearest thousandths. Homework Equations x2 + y2 = 1 The Attempt at a Solution x2+y2 = 1 loga2+logb2 =1 2loga+2logb = 1 2(loga+logb) = 1 loga + log b = 0.5 logb = 0.5−loga now i try...
16. ### B Exploring the Applications of Trigonometric Functions Beyond 90°

Why trigonometric functions are defined for unit circle, here "why" refers to what made them to define it this way, they may have defined it for right triangle only , can you give me a application where sin(120°) or sin, cos , tan of more than 90° is used to find some values like in physics or...
17. ### Complex numbers on the unit circle

Homework Statement Let ##z_1,z_2,z_3## be three complex numbers that lie on the unit circle in the complex plane, and ##z_1+z_2+z_3=0##. Show that the numbers are located at the vertices of an equilateral triangle. Homework EquationsThe Attempt at a Solution As far as I understand, I need to...
18. ### I Why trigonometric ratios were defined for a unit circle

To make it useful for any angles. I need a good explanation for this.
19. ### B Unit Circle and Other Trig Questions

When creating a right triangle in a unit circle how do you know where to place the leg from the terminal side? My textbook and Khan academy don't really explain this and it's just sort of assumed that I'd know. For example, If theta is equal to 135 degrees, where does the leg to complete the...
20. ### Want to talk about unit circle

Hi all, I was wandering on the web to collect some solid and justifiable reasons to answer a question "Why we always choose a unit circle."? I saw several websites and meanwhile I saw this post https://www.physicsforums.com/threads/trig-unit-circle-why.475575/. I saved it. But overall, I...
21. ### Question About Unit Circle (CircularFunction) of a Trig Func

Please take a look below example (the attached image below). How do I know that the angle ##\sin (\frac{7π}{4})## is corresponds to the coordinates ##(\frac{\sqrt {2}}{2}, -\frac{\sqrt{2}}{2})##? I know that ##\frac{7π}{4}## is 315°.

I get the dynamics of what the question is asking. What I don't get is how to know what pi/8 is. What I've been doing this entire time is plugging it into the calculator as a fraction, and getting the decimal value of it and comparing it to other values I know. I know that pi radians = 180...
23. ### Unit circle derived distribution

Hi Assume an x-coordinate from the unit circle is picked from a uniform distribution. This is the outcome of the random variable X with probability density function Xden(x) = 0.5 (-1 < x < 1). The random variable Y is related to the random variable X by Y = f(X) = √(1-X2) and X = g(Y) =...
24. ### Integrating over the unit circle

Homework Statement Suppose that the function ##f(z)## is analytic and that ##|f(z)| \le 1## for all ##|z| = 1##. Homework EquationsThe Attempt at a Solution I was hoping someone could verify my work. Okay, if I understand correctly, ##|f(z)| \le 1## is true for all all complex numbers ##z##...
25. ### A position versus time graph involving the unit circle

Homework Statement Homework Equations v(x)t= -wAsin(w+o) The Attempt at a Solution -5(10)sin(0)= 0 D is this the right procedure?
26. ### Unit Circle and Quantum Predictions [hidden variable model agrees with QM?]

I would like to reopen a discussion on assertions made by David Mermin such as: “There is no conceivable way to assign such instruction sets to the particles from one run to the next that can account for the fact that in all runs taken together, without regard to how the switches are set, the...
27. ### How to estimate 2 points on the unit circle

I have some noisy data (x-y coordinates) containing two distinct clusters of data. Each cluster is centred at an unknown point on the unit circle. How can I estimate these two points (green points in the diagram)? We can assume the noise is Gaussian and noise power is equal in x and y...
28. ### Integrate complex function over unit circle

Homework Statement Calculate ##\int _Kz^2exp(\frac{2}{z})dz## where ##K## is unit circle.Homework Equations The Attempt at a Solution Hmmm, I am having some troubles here. Here is how I tried: In general ##\int _\gamma f(z)dz=2\pi i\sum_{k=1}^{n}I(\gamma,a_k)Res(f,a_k)## where in my case...
29. ### Why does Sin represent Y on unit circle

As the title inquires, I am curious as to how or why the Sin function represents y coordinate on the unit circle.
30. ### Bivariate density on a unit circle

Homework Statement Consider the bivariate density of X and Y, f(x, y) = pi/2 for x^2 + y^2 ≤1 and y > x and = 0 otherwise. (a) Verify that this is a bivariate density (that is, the total volume ∫∫ f(x,y)dxdy = 1) Homework Equations The Attempt at a Solution The problem I'm having is...
31. ### Determining whether the unit circle group is a cyclic group

1. Homework Statement Let S be the set of complex numbers z such that |z|=1. Is S a cyclic group? 3. The Attempt at a Solution I think this group isn't cyclic but I don't know how to prove it. My only idea is: If G is a cyclic group, then there is an element x in G such that...
32. ### Homeomorphism of Unit Circle and XxX Product Space

Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane
33. ### Intuition why area of a period of sinx =4 = area of square unit circle

Homework Statement This isn't really homework, but I've been reviewing calc & trig and realized that the area of one period of sin(x) = 4. Since sin(θ) can be understood as the y-value of points along a unit circle, I noticed that the area of a unit square that bounds the unit circle is...
34. ### Contour integral of e^(-1/z) around a unit circle?

Homework Statement What is the integral of e-1/z around a unit circle centered at z = 0? Homework Equations - The Attempt at a Solution The Laurent expansion of this function gives : 1 - 1/z + 1/(2 z^2) - 1/(3! z^3) + . . . . . The residue of the pole inside is -1. So the integral...
35. ### Lattice on the closed unit circle?

Would either or both of these work as a lattice on the closed unit circle in the plane? (1) Using a linear order: Expressing points in polar coordinates (with angles 0≤θ<2π), define: (r,α) < (s,β) iff r<s or (r=s & α<β) (r,α) ≤ (s,β) iff (r,α) < (s,β) or (r=s & α=β) The meet and join...
36. ### Uniform convergence for heat kernel on unit circle

Homework Statement I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow. H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} . Homework Equations The Attempt at a...
37. ### MHB The Unit Circle, the Sinusoidal Curve, and the Slinky....

I seem to recall when taking college Trigonometry my professor saying that the unit circle and sinusoidal curves were basically a mathematical represention of a slinky in that the unit circle was the view of a slinky head on, so that what you saw in the two dimensional sense was a circle, and...
38. ### A question on degrees of maps of the fundamental group of the unit circle

Hello, I'm reading a textbook and in the textbook we are discussing the fundamental group of the unit circle and having some difficulty making out what a degree of a map is and why when there is a homotopy between two continuous maps f,g from S^{1} to S^{1} why the deg(f)=deg(g) We have...
39. ### Finding Points of Tangency for the Unit Circle

Hi I'm trying to study over break, this isn't an exact quote but its the part of the problem I need help with. Thanks. Homework Statement Draw the unit circle and plot the point P=(3,2). Observe there are TWO lines tangent to the circle passing through the point P. Lines L1 and L2 are...
40. ### How Does the Value of 'a' Affect the Integral on the Unit Circle Using Residues?

Homework Statement Calculate the integral ∫dθ/(1+acos(θ)) from 0 to 2∏ using residues. Homework Equations Res\underline{zo}(z)=lim\underline{z->zo} (z-z0)f(zo)*2∏i The Attempt at a Solution To start I sub cos(θ)=1/2(e^(iθ)+e^(-iθ)) so that de^(iθ)=ie^(iθ)dθ Re-writing in...
41. ### Showing a polynomial has at least one zero outside the unit circle.

The first thing that we should notice is that the leading coefficient $a_n = 1$. I was thinking about considering the factored form of p. I googled, and there is an algorithm called the "Schur-Cohn Algorithm" that is suppose to answer exactly this, but I can't find any information on it or...
42. ### Prove roots lie inside the unit circle

Homework Statement Let P(z)=1+2z+3z^2+...nz^(n-1). By considering (1-z)P(z) show that all the zeros of P(z) are inside the unit disk Homework Equations None given.. The Attempt at a Solution Well (1-z)P(z) = 1+z+z^2+...+nz^n and to find roots I set it to 0: 1+z+z^2+...+nz^n = 0...
43. ### Uniform Distribution on unit Circle

I keep reading that a random vector (X, Y) uniformly distributed over the unit circle has probability density \frac{1}{\pi}. The only proof I've seen is that f_{X,Y}(x,y) = \begin{cases} c, &\text{if }x^2 + y^2 \leq 1 \\ 0 &\text{otherwise}\end{cases} And then you solve for c by integrating...
44. ### MATLAB Plot unit circle in chebychev metric in MATLAB

Ok, so I'm trying to plot the unit circle using the chebyvhev metric, which should give me a square. I am trying this in MATLAB, using the 'pdist' and 'cmdscale' functions. My uber-complex code is the following: clc;clf;clear all; boundaryPlot=1.5; % Euclidean unit circle for i=1:360...
45. ### Mapping unit circle from one complex plane to another

I want to show that if the complex variables ζ and z and related via the relation z = (2/ζ) + ζ then the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane. Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y? Any help would be...
46. ### Prove Cone over Unit Circle Homeomorphic to Closed Unit Disc

Homework Statement This question comes out of "Introduction to Topology" by Mendelson, from the section on Identification Topologies. Let D be the closed unit disc in R^2, so that the boundary, S, is the unit circle. Let C=S\times [0,1], and A=S \times \{1\} \subset C. Prove that...
47. ### Why does the Unit Circle work?

In my Trig class, we learned about the unit circle and its relationship to the various trig functions (sin, cos, etc.). I am curious to know why the unit circle works the way it does, and the how it was "derived" so to speak. More specifically, why does radius of the circle have to be 1 for...
48. ### Why is integral of 1/z over unit circle not zero?

Ok I can do the integral and see that it is equal to 2∏i, but thinking about it in terms of 'adding up' all the points along the curve I feel like every every point gets canceled out by its antipode, e.g. 1/i and -1/i.
49. ### How to calculate arc length in unit circle

http://www.up98.org/upload/server1/01/z/cllb59cvnwaigmmar6b5.jpeg What is the method of calculating arc length in In the image above . x & y is known Thanks .
50. ### Identifying equivalence classes with the unit circle

Homework Statement Define a relation on R as follows. For a,b ∈ R, a ∼ b if a−b ∈ Z. Prove that this is an equivalence relation. Can you identify the set of equivalence classes with the unit circle in a natural way? Homework Equations The Attempt at a Solution I have already proven that this...