Trigonometry by Hand
Date: 05/24/2002 at 15:07:15 From: Sam Ellos Subject: Trigonometry by Hand Hi, My name is Sam Ellos. I am the type of person who just has to know how everything works. I have been wondering for months how to do sines, cosines, tangents, arcsines, arccosines, and arctangents without using tables or a calculator. I have tried to figure out these formulas for months, but I cannot get anything to work. If you could tell me these formulas, I would be very pleased.
Date: 05/25/2002 at 20:50:54 From: Doctor Rob Subject: Re: Trigonometry by Hand Thanks for writing to Ask Dr. Math, Sam! For sin(x) and cos(x), where x is in radians, use the infinite series sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ... cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ... Take as many terms as you need to get the accuracy you desire. Then tan(x) = sin(x)/cos(x). For arctan(y), if |y| < 1, use the infinite series arctan(y) = y - y^3/3 + y^5/5 - y^7/7 + y^9/9 - ... which gives you the angle in radians, again. If |y| > 1, then |1/y| < 1, and you can use the identity arctan(y) + arctan(1/y) = Pi/2. If |y| = 1, you can write down the answer by inspection. For arcsin(y) and arccos(y) with 0 < y < 1, you can use the facts that arcsin(y) = arctan(y/sqrt[1-y^2]), arccos(y) = arctan(sqrt[1-y^2]/y). If -1 < y < 0, then 0 < -y < 1, and there are the helpful identities arcsin(y) + arcsin(-y) = 0, arccos(y) + arccos(-y) = Pi. If y = -1, 0, or 1, you can write down the answer by inspection. The above series converge very rapidly for small values of x and y, but less so for large ones. You can ensure that the values of x you use in the first two series are between 0 and Pi/4 by applying various trigonometric identities relating sine and cosine values. Feel free to write again if I can help further. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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