**The trigonometric functions are defined in two related approaches:
(i) Right Triangle Trigonometry.
(ii) Unit Circle Trigonometry.
**

**(i) **In right triangle trigonometry, six
functions are defined with acute angles, right triangles, and the theory
of similar of triangles: sine, cosine, tangent, cotangent, secant and
cosecant.

In most courses on triangle trigonometry, the main attention is paid to
the sine, cosine, and tangent functions. These three functions are the
most frequently used for the study of triangles- especially with
problems of solving triangles using the laws of sines and cosines and
the use of tangents to measure inclination of an angle.

In the study of trigonometry applied to general triangles, treating
obtuse angles becomes relevant. The definitions of the trigonometric
functions are extended to angles of degree measures between 0 and
180. Making sense of the laws of sines and cosines and solving
triangles are central applications for this extension.

For our treatment of triangle trigonometry we consider all six functions as "core".

Note: The theory of similar right triangle shows that the value of any
one of these functions will determine the angle and thus the other five
functions, we could consider the sine function as the core of all
trigonometric functions for right triangles.

Details: Right Triangle Trigonometric Functions

**(ii)**
The unit circle approach to the trigonometric functions comes from
considering central angles in the geometry of circles- in particular
looking at a unit circle centered at the point $O=(0,0)$ in cartesian
coordinate plane.

Central angles with the positive horizontal axis as the initial
ray are determined by an arc length, $t$- the radian measure of the
angle. This arc also determines a right triangle by the center $O$, the
endpoint of the arc, $P(t)=(x(t), y(t)$, and the point on the horizontal
axis, $C(t)=(x(t),0)$. The right angle of the triangle is at the point
$C(t)$.

Since the triangle $OC(t)P(t)$ has a unit length for the measure of its
hypotenuse, the right triangle trigonometric functions of cosine and
sine correspond to the coordinates of $P(t)$. This correspondence is
used to extend the definition of these trigonometric functions to all
real numbers. The use of the coordinates related to the unit circle also
gives a sensible way to visualize the values of all the trigonometric
functions as lengths of line segments, with the easiest line segment
visualizations being connected to the sine, cosine, and tangent
functions. **
**Details: Unit Circle Trigonometric Functions