Trigonometric Functions Unit Circle

Unit circle:

In mathematics, a unit circle is a circle with a radius of one. Frequently, except 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 denoted as s1; the generalization to higher dimensions is the unit sphere.

Trigonometric functions on the unit circle:

All of the trigonometric functions of the angle θ can be modified by geometrically in terms of a unit circle centered at O.

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

cos(t) = x

sin(t) = y

The equation x2 + y2 = 1 gives the relation

cos2(t) + sin2(t) = 1

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

cos t = cos(2πk + t)

sin t = sin(2πk + t)

For any integer k

Area & Circumference:

1. Length of circumference:

The circumference’s length is related to the radius (r) by

c = 2*π*r

For unit circle circumference is 2*(because r = 1).

2. Area enclosed:

Area of the circle = π × area of the shaded square

Main article: Area of a disk

The area of the circle is π multiplied with the radius squared:

A = π *r2

For unit circle area is


Application:


Circle group:

The complex numbers can be pointed 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 has main applications in mathematics and science.

Application of the Unit Circle:

• It is used to understand the relationship between the unit circle and the trigonometric (sine and cosine) functions.

• Because of this reason it is used to solve a real-world application.