Unit circle, sine wave and cosine wave | Right triangle, ratios, functions | Trigonometry | Python

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The right triangle is the very foundation of trigonometry. The trigonometric ratios are ratios of two side lengths of a right triangle. Specifically, sine is the ratio of the opposite side to the hypotenuse; cosine is the ratio of the adjacent side to the hypotenuse; and tangent is the ratio of the opposite side to the adjacent side. However, the right triangle trigonometry is incomplete as it only addresses acute angles - a right triangle only has one right angle and two acute angles. To address all angles (i.e., acute angles, obtuse angles, reflex angles, and boundary angles), we need to switch to the unit circle trigonometry and redefine sine and cosine. The unit circle trigonometry is an extension of the right triangle trigonometry.

Because the radius of the unit circle is 1, we can deduce, based on the right triangle definitions of cosine and sine, that the cosine of the angle θ is the x-coordinate, and the sine of the angle θ is the y-coordinate of any point on the unit circle. These are the new definitions of sine and cosine, the unit circle definitions of sine and cosine, and they will enable us to find the sine and cosine of any angle. The angle θ is no longer limited to acute angles and, consequently, the trigonometric ratios defined in the right triangle become trigonometric functions whose domain is the real line.

Any circle is built up of an infinite series of right triangles. Thus, the unit circle is fundamentally linked to the right triangle.

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You can get the Python script that generates the animation from my Patreon page, namely   / sohcahtoa1609  .

Playlists
1) Geometry:    • Geometry  
2) Trigonometry:    • Trigonometry  
3) Arithmetic:    • Arithmetic  
4) Algebra:    • Algebra  
5) Differential calculus:    • Differential calculus  

6) Conic sections:    • Conic sections (geometry)  
7) Triangles:    • Triangles (geometry)  
8) Quadrilaterals:    • Quadrilaterals (geometry)  

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Lastly, a disclaimer: this video is meant to be a proof of concept.

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