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Cosmic Game © Douglass A. White, 2012 v151207 237 or cosine without infinitesimalizing to limits, and we have a clear understanding of how it relates to a tangent point on a curve and a constant velocity. This is so much simpler than the otherwise necessary division of 0 by 0 using the “ghosts of departed quantities” as Berkeley so aptly described them. The sketch below shows rotation by additional 90 degree steps into the third and fourth quadrant. The oscillating nature of sine and cosine are obvious. Sketch of an angle sum proof in trigonometry (See Wikipedia) Draw a horizontal line (the x-axis); mark an origin O. Draw a line from O at an angle above the horizontal line and a second line at an angle above that; the angle between the second line and the x-axis is . Place P on the line defined by at a unit distance from the origin. Let PQ be a line perpendicular to line defined by angle , drawn from point Q on this line to point P. OQP is a right angle. Let QA be a perpendicular from point A on the x-axis to Q and PB be a perpendicular from point B on the x-axis to P. OAQ and OBP are right angles. Draw QR parallel to the x-axis. Now angle (because , making
Cosmic Game © Douglass A. White, 2012 v151207 238 , and finally ) , so , so By substituting for and using Symmetry, we also get: Another simple "proof" can be given using Euler's formula known from complex analysis: Euler's formula is: Although it is more precise to say that Euler's formula entails the trigonometric identities, it follows that for angles and we have: Also using the following properties of exponential functions: Evaluating the product: Equating real and imaginary parts: Natural Logarithms The unique property of the natural logarithm base e, where y = e x is that de x /dx = e x . The result of that relation is that for every x value, the slope at that point is equal to the y value.
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