<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(*x*_{1},*y*_{1}) 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(*x*_{1},0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = *t*. Because PQ has length *y*_{1}, OQ length *x*_{1}, and OA length 1, sin(*t*) = *y*_{1} and cos(*t*) = *x*_{1}. Having established these equivalences, take another radius OR from the origin to a point R(−*x*_{1},*y*_{1}) on the circle such that the same angle *t* is formed with the negative arm of the *x*-axis. Now consider a point S(−*x*_{1},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

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(*x*_{1},*y*_{1}) 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(*x*_{1},0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = *t*. Because PQ has length *y*_{1}, OQ length *x*_{1}, and OA length 1, sin(*t*) = *y*_{1} and cos(*t*) = *x*_{1}. Having established these equivalences, take another radius OR from the origin to a point R(−*x*_{1},*y*_{1}) on the circle such that the same angle *t* is formed with the negative arm of the *x*-axis. Now consider a point S(−*x*_{1},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 (−*x*_{1},*y*_{1}) is the same as (cos(π − *t*),sin(π − *t*)) and (*x*_{1},*y*_{1}) 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*) = *y*_{1}/*x*_{1} and tan(π − *t*) = *y*_{1}/−*x*_{1}. 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 calculated without the use of a calculator by using the angle sum and difference formulas.

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

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

- $f_{0}(x)=x^{2}$

is a unit circle. It is a si

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

## Notes

**^** Confusingly, in geometry a unit circle is often considered to be a 2-sphere—not a 1-sphere. The unit circle is "embedded" in a 2-dimensional plane that contains both height and width—hence why it is called a 2-sphere in geometry. However, the surface of the circle itself is one-dimensional, which is why topologists classify it as a 1-sphere. For further discussion, see the technical distinction between a circle and a disk.^{[2]}