<P> 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 . </P> <P> 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 . </P> <P> 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 . </P> <P> Julia set of discrete nonlinear dynamical system with evolution function: </P>

The long length on a unit circle is