# Trigonometric functions of any angle

*Unit circle. Counting of angles in a unit circle.*

Negative and positive angles. Quarters of a unit circle.

Sine and cosine lines. Sine. Cosine. Signs of sine and

cosine in different quarters of a unit circle. Tangent

and cotangent lines. Tangent. Cotangent. Signs of

tangent and cotangent in different quarters of a unit

circle. Secant and cosecant.

Negative and positive angles. Quarters of a unit circle.

Sine and cosine lines. Sine. Cosine. Signs of sine and

cosine in different quarters of a unit circle. Tangent

and cotangent lines. Tangent. Cotangent. Signs of

tangent and cotangent in different quarters of a unit

circle. Secant and cosecant.

To build all trigonometry, laws of which would be valid for any angles ( not only for acute angles, but also for obtuse, positive and negative angles), it is

necessary to consider so called a *unit circle*, that is a circle with a radius, equal to 1 ( Fig.3 ).

Let draw two diameters: a horizontal AA’ and a vertical BB’. We count angles off a point A (starting point). Negative angles are counted in a clockwise,

positive in an opposite direction. A movable radius OC forms angle with

an immovable radius OA. It can be placed in the 1-st quarter ( COA ), in the 2-nd quarter ( DOA ), in the 3-rd quarter ( EOA ) or in the

4-th quarter ( FOA ). Considering OA and OB as positive directions and OA’ and OB’ as negative ones,

we determine trigonometric functions of angles as follows.

A *sine line *of an angle ( Fig.4 ) is a *vertical* diameter of a unit circle, a *cosine *

*line *of an angle – a *horizontal* diameter of a unit circle. A *sine* of an angle

( Fig.4 ) is the segment OB of a sine line, that is a projection of a movable radius

OK to a sine line; a *cosine* of an angle – the segment OA of a cosine line, that

is a projection of a movable radius OK to a cosine line.

Signs of sine and cosine in different quarters of a unit circle are shown on Fig.5 and Fig.6.

A *tangent line* ( Fig.7 ) is a tangent, drawn to a unit circle through the point A of a *horizontal* diameter.

A *cotangent line* ( Fig.8 ) is a tangent, drawn to a unit circle through the point B of a *vertical* diameter.

A *tangent* is a segment of a tangent line between the tangency point A and an intersection point ( D, E, etc., Fig.7 ) of a tangent line and a radius line.

A *cotangent* is a segment of a cotangent line between the tangency point B and an intersection point ( P, Q, etc., Fig.8 ) of a cotangent line and a radius line.

Signs of tangent and cotangent in different quarters of a unit circle see on Fig.9.

Secant and cosecant are determined as reciprocal values of cosine and sine correspondingly.