Computing Sine and Cosine
Date: 8/27/95 at 9:0:54 Subject: Sine/Cosine From: Bj\xrn St\frk Date: Sun, 27 Aug 1995 13:15:09 +0100 Dear Dr. Math Is it possible to compute sine and cosine for a given angle without using a calculator or a sine table?
Date: 8/27/95 at 17:13:53 From: Doctor Ken Subject: Re: Sine/Cosine Hello! Well, it depends. Sine and Cosine are defined in terms of geometric objects, so if you wanted to you could draw a circle of radius r on a pair of axes, lay your angle down on the circle, find where it intersects the circle, and if the coordinates of this point are (x,y) then the cosine of the angle is x/r and the sine is y/r. But I gather that you're looking for a more algebraic, formulaic way? Well, there's no algebraic way to compute exactly the trigonometric values of every angle, only certain ones like Pi/2, Pi/3, Pi/4, and things like that. You can, however, use an algebraic formula to find an _approximation_ to any angle. To approximate Sine, use the formula x^3 x^5 x^7 x - ---- + ----- - ----- + .... 3! 5! 7! For cosine, use the formula x^2 x^4 x^6 1 - ---- + ----- - ----- + .... 2! 4! 6! These both converge pretty quickly to the right value, and they converge most quickly when x isn't too far away from 0. -Doctor Ken, The Geometry Forum Check out our web site! http://mathforum.org/dr.math/dr-math.html
Date: 8/29/95 at 12:22:13 Subject: Sine/Cosine From: Bj\xrn St\frk Date: Tue, 29 Aug 1995 17:28:41 +0100 Hello again! :) Thank you, very much. But I don't exactly see where to use the angle in this formula. Say that I have an angle of 45 degrees, (or the same value in radians). How do I calculate sine/cosine for this angle?
Date: 8/30/95 at 11:12:16 From: Doctor Ken Subject: Re: Sine/Cosine Bjorn- The way you'd use these formulas is to plug in the angle (in radians) into the formula. So for the angle .25 radians, you'd get .25^3 .25^5 .25^7 .25 - ------ + ------- - ------- = .247404... 3! 5! 7! Keep in mind that this formula is only very accurate from about -Pi/2 to Pi/2. So to do an angle outside this range, use instead an angle that would have the same sine: instead of Sin[2.7] use Sin[Pi-2.7]. Do a similar thing for cosine, for which the formula is also good between -Pi/2 and Pi/2: instead of Cos[2.7] use -Cos[Pi-2.7]. -Doctor Ken, The Geometry Forum
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