Unit circle Definition and 106 Threads

Homework Statement Determine the behavior of convergence on the unit circle, ie |z| = 1 of: Ʃ \frac{z^{n}}{n^{2}(1 - z^{n})} Homework Equations Obviously this is divergent then z is a root of unity. The question is what happens when z is not a root of unity. The Attempt at a...

Consider the sequence (n) n=1 to infinity. Plot the sequence on the unit circle: n modulo 2*pi for n≥1. What do you see? Attempt: I really honestly have no idea what to do. We are learning in class about limit laws and how to prove them, so this question seems to be coming out of nowhere. :(

Homework Statement Let z1; z2; z3; z4 be four complex numbers on the unit circle (i.e., |z1|=|z2|=|z3|=|z4|=1). It is known that z1+z2+z3+z4=1+i . Find the value of 1/z1 + 1/z2 + 1/z3 + 1/z4 Homework Equations 1/z = barZ/|z|^2 The Attempt at a Solution I've been trying for about a day now...

I am learning about the unit circle and I am a bit confused. So, I have my circle drawn with radius 1 and I sketched a right angled triangle inside it so that the hypothenuse has a length of 1. I think what is making me confused is the meaning of sine, cosine and tangent. They are...

1. Show that S:= {(x,y)an element of R^2 : x^2 + y^2 =1} is connected. 2. Relevant theorems 1. Path-connected implies connected. The Attempt at a Solution Define f: [0,2pi] --> R^2 by f(x) = (cos(x),sin(x)). This map is continuous, and its image is S^1. The interval [0,2pi]...

Ok- I am teaching trigonometry to low level students right now and I am trying to figure out why they need to know the unit circle. Are there some interesting things they can learn about by using a unit circle? So far, we pretended it was a magic-barbie-sized-half-underground-ferris-wheel...

Hi, I have been told that in R^2 the unit circle {(x,y) | x^2 + y^2 = 1} is smoothly mappable to the curve {(x,y) | x^4 + y^2 = 1}. Can someone please tell me what this smooth map is between them? I can only think of using the map (x,y) --> (sqrt(x), y) if x is non-negative and (sqrt(-x), y)...

The book talks about a unit circle... 360 deg = 2(pi) rad if it wasnt a unit circle... say r = 4 would it then be 360 deg = 8(pi) rad?

Homework Statement consider the family of complex mappings: z -> Ma(z) = (z-a)/(áz-1) (a constant) (á is complex conjugate of a) Show that Ma(z) maps the unit circle to itself.Homework Equations the solution should look something like this i guess: Ma(ei*alpha) = ei*alpha The Attempt at a...

Homework Statement Choose a point in the unit axis, say x.Let Y be the distance of that point and the point where thε perpendicular line crosses the unit circle. Find the density and cumulative functions of Y. Homework Equations Basic trigonometry i guess. The Attempt at a Solution...

Does anyone know the (x,y) solutions on the unit circle for 15, 75, 105, 165, 195, 255, 285, or 345 degrees?

I am having a real tough time memorizing the unit circle and it's values. What would you suggest to make easier for me to remember the quadrants, square roots, and radians?

hello there hi everybody just i have been taken my final exam for calculus one CALCULUS I there was one qeustion which i was confouse while i was reading it Set up the intgeral area of unit circle?

Hi everyone Consider a 2x2 partitioned matrix as follow: A = [ B1 B2 ; B3 B4 ] I'm sure that all eigenvalues of A are on the unit circle (i.e., abs (all eig) = 1 ). but, I don't know how to prove it. Is there any theorem?

Homework Statement The area in the region inside the unit circle and above the graph of f(x) = x^5 Homework Equations I don't know how to type the equation in here but the area is the integral between two integration points of the higher curve minus the lower curve. The Attempt at a...

Homework Statement We are given the unit circle and the point (5,2). There are two lines that are tangent to the unit circle and they both intersect at the point (5,2). What are the points where these lines are tangent with the unit circle. Homework Equations Tangent line of a circle at...

Can somebody give me an example whereby I use the inversion with respect to a circle (unit circle or otherwise) and the problem becomes easier. I guess I am asking: how do I make use of this notion. Or a problem that involves inversion, period. Thank you

Do I have memorize the entire Unit circle ?? Homework Statement I am currently takin a trig class and I was a bit daunted by the Unit Circle and all its special angles. My question is... Do I haveto memorize the entire Unit Circle and its angles?? Will it be given to me during exams(as...

Homework Statement Find all solutions to the equation below such that -180° \leq x \leq 90° 2sin2x + sinx = 0Homework Equations The Attempt at a Solution 2sin2x + sinx = 0 sinx[2sinx + 1] = 0 sin x = 0 sinx= -1/2 x = {0° + 360°n {-180° + 360°n n\epsilonI

I am currently working on an implementation of a Symbolic Algebra system similar to existing products. In this system, I would like to be able to display the exact symbolic values of trigonometric functions for any given angle in radians. ex: sin(PI/6) = "1/2" My problem stems from...

Homework Statement Find all values of x in the interval [0,2pi] that satisfy the inequality. sin x > cos x Homework Equations Unit circle The Attempt at a Solution pi/3 > x > 7pi/6 Is that correct?

Homework Statement Express f(x,y) = 1/sqrt(x^2 + y^2) . (y/sqrt(x^2 + y^2)) .exp(-2sqrt(x^2 + y^2)) in terms of polar coordinates \rho and \varphi then evaluate the integral over a circle of radius 1, centered at the origin. Homework Equations x = \rhocos\varphi y =...

hopefully we all know x^2 + y^2 =1 and x=cost y=sint, t between 0 and 2pi. There's also one with slope; x= (1-t^2)/(1+t^2) y= (2t)/(1+t^2) I was wondering if this counts as a separate one x+iy=e^it, t also between 0 and 2pi or if this is analogous to the trig parameterization...

Homework Statement Define a relation on R as follows. Two real numbers x, y are equivalent if x - y \epsilon Z . Show that the set of equivalence classes of this relation is bijective to the set of points on the unit circle. Homework Equations N/A? I don't think there are any special...

Homework Statement Calculate the work W_{A B} done by the force F using Newton's laws (F=ma, etc), when a particle moves from the point A to the point B along the unit circle. The angle is \theta. No friction. How do you define kinetic energy in polar coordinates?Homework Equations...

Homework Statement sin3x=.966 then x could be equal to (answer in degrees) Homework Equations solving trigonometric equations The Attempt at a Solution sin3x=3sinx 3sinx=.966 sinx=.966/3 sinx=.322 cos^2x + sin^2x=1 (pythagorean theory) cos^2x +.322^2=1 1-.322^2=cos^2x lol i...

The question I had was to show that if a function is continuous, open and bijective then it is a homeomorphism. At first I said "no" because I thought of the example showing that [0,2pi) is not homeomorphic to the unit circle S. I knew that f(x)=(sinx,cosx) is a continuous bijection whose...

Given the double integral \int\int_R \sqrt{}x^2+y^2 dx dy where R is the unit circle. We are only given the equation for the unit circle but don't we need more equations so I can change the equations to a single variable and then find the Jacobian so how do I find the Jacobian. How do I find a...

For the double integral \int\int_R sqrt(x^2+y^2) dx dy where R is the unit circle. I got\int_0^\pi\int_1^1 sqrt(r2) r dr dtheta Then after the integration I got an answer of 2pi/3 as my final answer. Is this right. The bottom of the 2nd integral is -1 not 1

Homework Statement Show that the transformation _ __ _ __ | 0 -2 1 || x | | -2 2 0 || y | |_2 -2 1 _||_ 1_| takes all points on parabola y2=x onto the unit circle x2+y2=1 Homework Equations The Attempt at a Solution I can't find out what to do I just...

Homework Statement Hi all. Today I had to solve: \cos \theta = -1/2. What I did was to look in a table to find that \theta = 2\pi/3 \quad \text{and}\quad \theta = 4\pi/3. My question is what is the general strategy when I wish to write this as a a function of an integer n? Is there even a...

Let x^2_1+x^2_2=1 be an unit circle upon a finite field Z_{p} where p is a prime. Is there any algorithm (other than the brute force algorithm) which can give all the possible solutions (x_1,x_2)\in Z_{p}\times Z_{p} as well as the total number of such solutions? If exists, what is the...

I need some guidance into understanding Radian Measure and the Unit Circle. This was the topic where I tanked and had to drop the course. I'm going to pick it up again next fall and want to start preparing now. Any help is appreciated. Sean

Homework Statement What is the line integral of F(x,y,z) = (xy, x, xyz) over the unit circle c(t) = (cost, sint) t E (0,2pi) ? Homework Equations integral= (f(c(t))*c'(t))dt) The Attempt at a Solution Ok, so I tried solving this like I would any other line integral using the given...

Homework Statement I'm trying to do a few problems that ask me to "find the point (x,y) on the unit circle that corresponds to the real number t." Examples of these problems are: t = pi / 4 t = 7pi / 6 t = 4pi / 3 etc etc Homework Equations The Attempt at a Solution I...

I have a unit circle: x^2+y^2 <= 1 And I'm asked to convert it to a square with verticies (0,0),(0,1),(1,0),(1,1). Now obviously I have to do this in polar coordinates, so I've rewritten the equation as: x = cos t y = sin t I'm sort of drawing a blank after setting up these...

[SOLVED] Unit Circle How can I use the unit circle to get the right answer. I understand the 30 60 90 and 45 45 90 triangles, but come to a problem when using this method with cosine. For example cosine(3pi/2) is -[2^(1/2)]/2. Using my method I get the positive. Please explain the methods of...

Homework Statement The problem comes with a diagram but I'll use the wikipedia diagram because it's nice and pretty and I'll just rearrange the letters to suit it. http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg Just in case the image doesn't load in the page...

Just trying to find a way to work out the trig ratios for angles with large fracetions in the unit circle (e.g. sin(15pi/2) etc..) for angles with smaller fractions like cos(-7pi/4) i can solve easily like this: 7/4 = 1.75 = 45 degree (pi/4) angle in the 1st quadrant (because its negative)...

What would the domain of y = sqrt(cosx) be in mathematical terms. I know that it is all the reals that lie in the first and fourth quadrant of the unit circle, but how would you express that in mathematical terms?

I'd like to map the open unit circle to the open ellipse x/A^2 + y/B^2 = 1. How would I go about doing this? I really have no idea how to go about doing these mappings. I'm working with the text Complex Var. and Applications by Ward and Churchill which has a table of mappings in the back...

Homework Statement a sample problem: arcsin(-1/2) 2. The attempt at a solution do i look at the unit circle and find the y-coordinate or x-coordinate that has -1/2? i did ASTC, and figure that it'd be in either quad 3 or quad 4; to tell you the truth i don't understand how to use...

[SOLVED] unit circle Homework Statement My book contains the following problem: Let U be the multiplication group \{z \in C : |z| = 1\} 1) Let z_0 be in U. Show that U z_0 = \{ z z_0 : z \in U \} is a subgroup of U, and compute U mod U z_0. 2) To what group is U/<-1> isomorphic to...

Hi all, My math is kinda weak but I'm re-attempting a precalculus course . I was just wondering exactly how the unit circle helps me?? I mean,I can generate it quite easily(from memory,but)...but why not just convert to degrees and enter it into my calculator? Also,finding angles that...

Homework Statement Each of the following paths describes the motion of a particle having the same path, namely the unit circle x^2 + y^2 =1. Although the path for each particle is the same, the behavior of each particle is different. For each particle, answer the following questions: i. ...

a = 60*pi/180; a1 = (pi - a)/2; a2 = (pi + a)/2; theta = a1: a/60: a2; rho = ones(size(theta)); rho1 = rho*sin(a1)./sin(theta); polar(theta, rho); hold on; polar(theta, rho1) The above commands will draw a segment of a unit circle which starts from 60^{o} to 120^{o}. I know...

Homework Statement How to find the conditions on the coefficients of a quadratic equation for the roots to be outside the unit circle eg bx^2 + x - 1 = 0 where b is a constant How do we find the condition(s) that b must satisfy such that the roots of the quadratic lie outside the unit circle...

Prove that the unit circle, for an inner product on lR^2 is defined as the set of all vectors of unit length ||v|| = 1, of the non-standard inner product v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2 is an ellipse. I know that norm squared will be (v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2) (v_1...

This was an exam question I had at the end of 2005 in my uni entrance exams. Can you do it? Establish the inequality xcosx < sinx < x where 0<x<pi/2 using the unit circle. is the tex working?

So I tried actually calculating it and came up with a mess. So then I did the unit interval and got 1/3. So, by analogy, it's 1/3 for the unit circle? Any cute proofs for whatever the correct answer is?