Trigonometry
Trigonometric Functions
The unit circle or the trigonometric circle
Division of the circumference of the unit circle to the characteristic angles
Definitions of trigonometric functions
The unit circle or trigonometric circle
A circle of radius r = 1, with the center at the origin O(0, 0) of a coordinate system, we call the unit or trigonometric circle, see the figure below.
The arc of the unit circle that describes a point traveling anticlockwise (by convention, clockwise is taken to be negative direction) from the initial position P1(1, 0) on the x-axis, along the circumference, to the terminal position P equals the angular measure/distance x = arad, in radians.
 An angle is in standard position if its initial side lies along the positive x-axis. If we take the positive direction of the x-axis as the beginning of a measurement of an angle (i.e., a = 0rad, both sides of an angle lie on the x-axis), and the unit point P1 as the initial point of measuring the arc, then the terminal side of an angle, which passes through the terminal point P of the arc, rotating around the origin (in any direction) describes different angles, and the terminal point P corresponding arcs, x = arad + k · 2p,  k = 0, ±1, ±2, ±3, . . . or   x = a° + k · 360°,   k Î Z.
It means that every arc x ends in the same point P in which ends the corresponding arc a.
Thus, at any point P on the circumference of the unit circle end infinite arcs x = a + k · 2p, which differ by the multiplier 2p, and any number x associates only one point P.
Division of the circumference of the unit circle to the characteristic angles
There is a common division of the circumference of the unit circle to the characteristic angles or the corresponding arcs which are the multipliers of the angles, 30° (p/ 6) and 45° (p/ 4).
 We can say that a unit circle is at the same time numerical circle. The numerical circle shown in the right figure is formed by winding the positive part of number line, with the unit that equals the radius, around the unit circle in the anticlockwise direction and its negative part in clockwise direction. Therefore, the terms angle, arc and number in the trigonometric definitions and expressions are mutually interchangeable.

Example:   In which quadrant lies second or the terminal side of the angle x = 1280°
 Solution:  Dividing the given angle by 360° we calculate the number of rotations, or round angles, described by terminal side of the angle x, and the remaining angle a° position of which we want to find. since  x = a° + k · 360°  then   k = 3  and  a = 200°. therefore, terminal side of the angle x lies in the third quadrant.
Example:   In which quadrant lies the endpoint of the arc x = - 47p/3 of a unit circle.
 Solution:  Given arc
can be expanded to

Thus, the endpoint of the arc x moved around a unit circle in the clockwise (negative) direction 7 times and described additional arc a = - (5/3)prad, so its endpoint P lies in the first quadrant.
Definitions of trigonometric functions
Let x be an arc of the unit circle measured counterclockwise from the x-axis. It is at the same time the circular measure of the subtended central angle a as is shown in the below figure
 In accordance with the definitions of trigonometric functions in a right-angled triangle, - the sine of an angle a (sin a) in a right triangle is the ratio  of the side opposite the angle to the hypotenuse. - the cosine of an angle a (cos a) in a right triangle is the ratio of the side adjacent to it to the hypotenuse. Thus, from the right triangle OP′P, follows sin x = PP′    The sine of arc x is the ordinate of the arc endpoint. cos x = OP′  The cosine of arc x is the abscissa of the arc endpoint.
 The tangent of an angle a (tan a) in a right triangle is the ratio of the lengths of the opposite to the adjacent side. The cotangent is defined as reciprocal of the tangent, thus From the similarity of the triangles OP′P and OP1S1,
Hence, the definition of the tangent function in the unit circle,
tan x = P1S1    The tangent of an arc x is the ordinate of intersection of the second or terminal side (or its
extension) of the given angle and the tangent line x = 1.
 From the similarity of the triangles, OP′P and OP2S2,
cot x = P2S2    The cotangent of an arc x is the abscissa of intersection of the second or terminal side (or its
extension) of the given angle and the tangent  y = 1.
It is obvious from the definitions that the tangent function is not defined for arguments x for which cos x = 0,
as well as the cotangent function is not defined for the arguments for which
sin x = 0.
Functions contents B