Unit circle

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

$${\displaystyle z=e^{it}=\cos t+i\sin t=\operatorname {cis} (t)}$$

for all t (see also: cis). This relation represents Euler's formula. In quantum mechanics, this is referred to as the phase factor. It can also be defined as the set of complex numbers satisfying

$${\displaystyle |z|=1.}$$

Animation of the unit circle with angles

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)

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

$${\displaystyle \cos \theta =x\quad {\text{and}}\quad \sin \theta =y.}$$

The equation x² + y² = 1 gives the relation

$${\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}$$

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

$${\displaystyle \cos \theta =\cos(2\pi k+\theta )}$$

$${\displaystyle \sin \theta =\sin(2\pi k+\theta )}$$

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 OP from the origin O 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 OP has length 1 as a radius on the unit circle, 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(3π/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 easily calculated by hand using the angle sum and difference formulas.

The unit circle, showing coordinates of certain points

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 ${\displaystyle \mathbb {T} .}$ 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.

Unit circle in complex dynamics

The Julia set of discrete nonlinear dynamical system with evolution function:

$${\displaystyle f_{0}(x)=x^{2}}$$

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

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