Unit circle

Unit circle

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 Template:Math, the circumference of a unit circle is Template:Math.

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 Template:Math because it is a one-dimensional unit [[n-sphere|Template:Math-sphere]].Template:Refn

If Template:Math is a point on the unit circle's circumference, then Template:Math and Template:Math are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, Template:Math and Template:Math satisfy the equation $$x^2+y^2=1.$$

Since Template:Math for all Template:Math, and since the reflection of any point on the unit circle about the Template:Math- or Template:Math-axis is also on the unit circle, the above equation holds for all points Template:Math 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.

In the complex plane

File:Unitycircle-complex.gif

Animation of the unit circle with angles

In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers Template:Mvar such that $|z|=1.$ When broken into real and imaginary components $z=x+iy,$ this condition is $|z{|}^2=z\overline{z}=x^2+y^2=1.$

The complex unit circle can be parametrized by angle measure $\theta$ from the positive real axis using the complex exponential function, $z=e^{i\theta }=\cos \theta +i\sin \theta .$ (See Euler's formula.)

Under the complex multiplication operation, the unit complex numbers form a group called the circle group, usually denoted $𝕋.$ In quantum mechanics, a unit complex number is called a phase factor.

Trigonometric functions on the unit circle

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All of the trigonometric functions of the angle Template:Math (theta) can be constructed geometrically in terms of a unit circle centered at O.

File:Periodic sine.svg

Sine function on unit circle (top) and its graph (bottom)

The trigonometric functions cosine and sine of angle Template:Math may be defined on the unit circle as follows: If Template:Math is a point on the unit circle, and if the ray from the origin Template:Math to Template:Math makes an angle Template:Math from the positive Template:Math-axis, (where counterclockwise turning is positive), then $$\cos \theta =x\text{and}\sin \theta =y.$$

The equation Template:Math gives the relation $${\cos }^2\theta +{\sin }^2\theta =1.$$

The unit circle also demonstrates that sine and cosine are periodic functions, with the identities $$\cos \theta =\cos (2\pi k+\theta )$$ $$\sin \theta =\sin (2\pi k+\theta )$$ for any integer Template:Math.

Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius Template:Math from the origin Template:Math to a point Template:Math on the unit circle such that an angle Template:Math with Template:Math is formed with the positive arm of the Template:Math-axis. Now consider a point Template:Math and line segments Template:Math. The result is a right triangle Template:Math with Template:Math. Because Template:Math has length Template:Math, Template:Math length Template:Math, and Template:Math has length 1 as a radius on the unit circle, Template:Math and Template:Math. Having established these equivalences, take another radius Template:Math from the origin to a point Template:Math on the circle such that the same angle Template:Math is formed with the negative arm of the Template:Math-axis. Now consider a point Template:Math and line segments Template:Math. The result is a right triangle Template:Math with Template:Math. It can hence be seen that, because Template:Math, Template:Math is at Template:Math in the same way that P is at Template:Math. The conclusion is that, since Template:Math is the same as Template:Math and Template:Math is the same as Template:Math, it is true that Template:Math and Template:Math. It may be inferred in a similar manner that Template:Math, since Template:Math and Template:Math. A simple demonstration of the above can be seen in the equality Template:Math.

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than Template:Sfrac. However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2Template:Pi. 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 easily calculated by hand using the angle sum and difference formulas.

File:Unit circle angles color.svg

The unit circle, showing coordinates of certain points

Complex dynamics

File:Erays.svg

Unit circle in complex dynamics

The Julia set of discrete nonlinear dynamical system with evolution function: $$f_0(x)=x^2$$ is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems.

See also

• Angle measure
• Pythagorean trigonometric identity
• Riemannian circle
• Radian
• Unit disk
• Unit sphere
• Unit hyperbola
• Unit square
• Turn (angle)
• z-transform
• Smith chart