{"id":109,"date":"2020-12-02T00:01:54","date_gmt":"2020-12-02T00:01:54","guid":{"rendered":"https:\/\/mathdynamics.net\/blog\/?p=109"},"modified":"2021-10-28T21:32:08","modified_gmt":"2021-10-28T21:32:08","slug":"a-treatise-on-trigonometry","status":"publish","type":"post","link":"https:\/\/mathdynamics.net\/blog\/index.php\/2020\/12\/02\/a-treatise-on-trigonometry\/","title":{"rendered":"A Treatise on Trigonometry"},"content":{"rendered":"

Trigonometry is based on the fundamental principle of ONE length.\u00a0 The length in this context is immaterial because with Trigonometry we only need be concerned about ONE.\u00a0 For example if you draw a circle it is ONE circle no matter how big or small.\u00a0 Because it is a circle it has only ONE center with a constant distance to the edge of the circle.\u00a0 This constant distance is ONE and is called the Radius.<\/p>\n

\"The
The Unit Circle is fundamental to Trigonometry. Any circle has a constant Circumference that is exactly 3.14159… times its Diameter. The “units”, i.e. inches, centimeters, … are immaterial. Its radius is simply ONE unit.<\/figcaption><\/figure>\n

Simple so far, just a circle with a center and a radius of ONE.\u00a0 Now consider the radius as it sweeps through a small region of the circle forming a triangle with two sides of an equal length ONE\u00a0 and forming an angle at the apex of the triangle.\u00a0 An Angle is a unit that specifies the amount of the sweep of the radius that formed the triangle.<\/p>\n

At this point we are finished with the circle.\u00a0 It is no longer needed.\u00a0 Also, our Triangle can now be moved and reoriented any way we wish.\u00a0 \"\"It is imperative though that the lengths of the two radii forming the triangle remain equal and ONE.\u00a0 The angle is most visually apparent when one of the sides of the Triangle lies along a horizontal axis.\u00a0 At this point a perpendicular line can be drawn between the tip of one of radii and intersecting the other radii.\u00a0 The perpendicular line intersects the radii\u00a0 such that the length between the apex (angle)\u00a0 and the intersection is a ratio of ONE.<\/p>\n

Now, this is where it gets complicated.\u00a0 Since the triangle has been conveniently aligned with one of the radii along the horizontal axis it can be said that the SIN is the distance (as a percentage of ONE) along a corresponding VERTICAL AXIS between the TIP of one radii and a HORIZONTAL AXIS.\u00a0 Likewise the COS is the distance along a HORIZONTAL AXIS\u00a0 between the TIP of one radii and a VERTICAL AXIS.<\/p>\n