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From Wikipedia, the free encyclopedia

Illustration of a unit circle. The variable t is an angle measure.
Illustration of a unit circle. The variable t is an angle measure.
Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. Since C = 2πr, the circumference of a unit circle is 2π.
Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. Since C = 2πr, the circumference of a unit circle is .

In mathematics, a unit circle is a circle with unit radius. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit 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

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.

YouTube Encyclopedic

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  • ✪ Introduction to the unit circle | Trigonometry | Khan Academy
  • ✪ ❖ A Way to remember the Entire Unit Circle for Trigonometry ❖
  • ✪ Why you should never memorize the unit circle How to use without memorizing

Transcription

What I have attempted to draw here is a unit circle. And the fact I'm calling it a unit circle means it has a radius of 1. So this length from the center-- and I centered it at the origin-- this length, from the center to any point on the circle, is of length 1. So what would this coordinate be right over there, right where it intersects along the x-axis? Well, x would be 1, y would be 0. What would this coordinate be up here? Well, we've gone 1 above the origin, but we haven't moved to the left or the right. So our x value is 0. Our y value is 1. What about back here? Well, here our x value is 1. We've moved 1 to the left. And we haven't moved up or down, so our y value is 0. And what about down here? Well, we've gone a unit down, or 1 below the origin. But we haven't moved in the xy direction. So our x is 0, and our y is negative 1. Now, with that out of the way, I'm going to draw an angle. And the way I'm going to draw this angle-- I'm going to define a convention for positive angles. I'm going to say a positive angle-- well, the initial side of the angle we're always going to do along the positive x-axis. So you can kind of view it as the starting side, the initial side of an angle. And then to draw a positive angle, the terminal side, we're going to move in a counterclockwise direction. So positive angle means we're going counterclockwise. And this is just the convention I'm going to use, and it's also the convention that is typically used. And so you can imagine a negative angle would move in a clockwise direction. So let me draw a positive angle. So a positive angle might look something like this. This is the initial side. And then from that, I go in a counterclockwise direction until I measure out the angle. And then this is the terminal side. So this is a positive angle theta. And what I want to do is think about this point of intersection between the terminal side of this angle and my unit circle. And let's just say it has the coordinates a comma b. The x value where it intersects is a. The y value where it intersects is b. And the whole point of what I'm doing here is I'm going to see how this unit circle might be able to help us extend our traditional definitions of trig functions. And so what I want to do is I want to make this theta part of a right triangle. So to make it part of a right triangle, let me drop an altitude right over here. And let me make it clear that this is a 90-degree angle. So this theta is part of this right triangle. So let's see what we can figure out about the sides of this right triangle. So the first question I have to ask you is, what is the length of the hypotenuse of this right triangle that I have just constructed? Well, this hypotenuse is just a radius of a unit circle. The unit circle has a radius of 1. So the hypotenuse has length 1. Now, what is the length of this blue side right over here? You could view this as the opposite side to the angle. Well, this height is the exact same thing as the y-coordinate of this point of intersection. So this height right over here is going to be equal to b. The y-coordinate right over here is b. This height is equal to b. Now, exact same logic-- what is the length of this base going to be? The base just of the right triangle? Well, this is going to be the x-coordinate of this point of intersection. If you were to drop this down, this is the point x is equal to a. Or this whole length between the origin and that is of length a. Now that we have set that up, what is the cosine-- let me use the same green-- what is the cosine of my angle going to be in terms of a's and b's and any other numbers that might show up? Well, to think about that, we just need our soh cah toa definition. That's the only one we have now. We are actually in the process of extending it-- soh cah toa definition of trig functions. And the cah part is what helps us with cosine. It tells us that the cosine of an angle is equal to the length of the adjacent side over the hypotenuse. So what's this going to be? The length of the adjacent side-- for this angle, the adjacent side has length a. So it's going to be equal to a over-- what's the length of the hypotenuse? Well, that's just 1. So the cosine of theta is just equal to a. Let me write this down again. So the cosine of theta is just equal to a. It's equal to the x-coordinate of where this terminal side of the angle intersected the unit circle. Now let's think about the sine of theta. And I'm going to do it in-- let me see-- I'll do it in orange. So what's the sine of theta going to be? Well, we just have to look at the soh part of our soh cah toa definition. It tells us that sine is opposite over hypotenuse. Well, the opposite side here has length b. And the hypotenuse has length 1. So our sine of theta is equal to b. So an interesting thing-- this coordinate, this point where our terminal side of our angle intersected the unit circle, that point a, b-- we could also view this as a is the same thing as cosine of theta. And b is the same thing as sine of theta. Well, that's interesting. We just used our soh cah toa definition. Now, can we in some way use this to extend soh cah toa? Because soh cah toa has a problem. It works out fine if our angle is greater than 0 degrees, if we're dealing with degrees, and if it's less than 90 degrees. We can always make it part of a right triangle. But soh cah toa starts to break down as our angle is either 0 or maybe even becomes negative, or as our angle is 90 degrees or more. You can't have a right triangle with two 90-degree angles in it. It starts to break down. Let me make this clear. So sure, this is a right triangle, so the angle is pretty large. I can make the angle even larger and still have a right triangle. Even larger-- but I can never get quite to 90 degrees. At 90 degrees, it's not clear that I have a right triangle any more. It all seems to break down. And especially the case, what happens when I go beyond 90 degrees. So let's see if we can use what we said up here. Let's set up a new definition of our trig functions which is really an extension of soh cah toa and is consistent with soh cah toa. Instead of defining cosine as if I have a right triangle, and saying, OK, it's the adjacent over the hypotenuse. Sine is the opposite over the hypotenuse. Tangent is opposite over adjacent. Why don't I just say, for any angle, I can draw it in the unit circle using this convention that I just set up? And let's just say that the cosine of our angle is equal to the x-coordinate where we intersect, where the terminal side of our angle intersects the unit circle. And why don't we define sine of theta to be equal to the y-coordinate where the terminal side of the angle intersects the unit circle? So essentially, for any angle, this point is going to define cosine of theta and sine of theta. And so what would be a reasonable definition for tangent of theta? Well, tangent of theta-- even with soh cah toa-- could be defined as sine of theta over cosine of theta, which in this case is just going to be the y-coordinate where we intersect the unit circle over the x-coordinate. In the next few videos, I'll show some examples where we use the unit circle definition to start evaluating some trig ratios.

Contents

In the complex plane

The unit circle can be considered as the unit complex numbers, i.e., the set of complex numbers z of the form

for all t (see also: cis). This relation represents Euler's formula. In quantum mechanics, this is referred to as phase factor.

Animation of the unit circle with angles(Click to view)
Animation of the unit circle with angles(Click to view)

Trigonometric functions on the unit circle

All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O.
All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O.
Sine function on unit circle (top) and its graph (bottom)
Sine function on unit circle (top) and its graph (bottom)

The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then

The equation x2 + y2 = 1 gives the relation

The unit circle also demonstrates that sine and cosine are periodic functions, with the identities

for any integer k.

Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P(x1,y1) on the unit circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q(x1,0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin(t) = y1 and cos(t) = x1. Having established these equivalences, take another radius OR from the origin to a point R(−x1,y1) on the circle such that the same angle t is formed with the negative arm of the x-axis. Now consider a point S(−x1,0) and line segments RS ⊥ OS. The result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at (cos(π − t),sin(π − t)) in the same way that P is at (cos(t),sin(t)). The conclusion is that, since (−x1,y1) is the same as (cos(π − t),sin(π − t)) and (x1,y1) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(π − t) and −cos(t) = cos(π − t). It may be inferred in a similar manner that tan(π − t) = −tan(t), since tan(t) = y1/x1 and tan(π − t) = y1/x1. A simple demonstration of the above can be seen in the equality sin(π/4) = sin(/4) = 1/2.

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right.

Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the angle sum and difference formulas.

The unit circle, showing coordinates of certain points
The unit circle, showing coordinates of certain points

Circle group

Complex numbers can be identified with points in the Euclidean plane, namely the number a + bi is identified with the point (a, b). Under this identification, the unit circle is a group under multiplication, called the circle group; it is usually denoted 𝕋. On the plane, multiplication by cos θ + i sin θ gives a counterclockwise rotation by θ. This group has important applications in mathematics and science.[example  needed]

Complex dynamics

Unit circle in complex dynamics
Unit circle in complex dynamics

Julia set of discrete nonlinear dynamical system with evolution function:

is a unit circle. It is a simplest case so it is widely used in study of dynamical systems.

See also

External links

  • Weisstein, Eric W. "Unit circle". MathWorld.
This page was last edited on 19 May 2019, at 23:29
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