Definitions:

Original trigonometric definitions relate to right-angled triangle, whereas the one with a unit circle is the extended definitions.

Click to enlarge.

The above is the original definitions of trigonometric functions.

Extended definitions make the functions still defined when the angle is larger than or equal to a right angle.

Click to enlarge.

For more intelligent readers, you will probably notice that sine function ranges from -1 to 1, cosine function ranges from -1 to 1 (because the coordinates are all in these ranges). However, the amplitude of the tangent function is infinity because x-coordinate can be 0.

Please read The Trigonometric Functions here.

Another very good deduction using Pythagorean theorem related to the trigonometric functions:

which is a basic identity in Trigonometry. Later on you will see other more complicated ones.