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

Original Definition

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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.

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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.