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/