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Tangent without a Calculator

Date: 06/24/2001 at 05:12:20
From: martin
Subject: Calculus

How do I find a tangent value if I have only an angle, and can't use a 
calculator? I've tried taking the derivative, but that doesn't seem to 
work and requires a calculator besides.

Date: 06/27/2001 at 16:56:23
From: Doctor Rob
Subject: Re: Calculus

Thanks for writing to Ask Dr. Math, Martin.

First convert the angle to radians if it is not already in that form.
Let the result be x. Then use the following series to compute the
cosine of x:

   cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...

Use as many terms as you need to get the accuracy you want (that is,
until the first omitted term is less than your accuracy tolerance).
Then use the identity

   tan^2(x) = 1/cos^2(x) - 1

to find tan^2(x). Then take the square root (affixing the sign
appropriate to the quadrant in which x falls) to find tan(x).

Example: Suppose you want to find the tangent of 20 degrees, or
x = Pi/9 radians to the nearest millionth. We'll keep two extra digits 
as a safety margin for round-off error.  Then

   x = 3.14195265.../9 = 0.34906585...,
   x^2 = 0.12184697...,
   1                        = 1,
   x^2/2! =      (1)*x^2/2  = 0.06092348...,
   x^4/4! = (x^2/2!)*x^2/12 = 0.00061861...,
   x^6/6! = (x^4/4!)*x^2/30 = 0.00000251...,
   x^8/8! = (x^6/6!)*x^2/56 = 0.00000001...,
   cos(x) = 1 - 0.06092348 + 0.00061861 - 0.00000251 + 0.00000001,
          = 0.93969263...,
   tan^2(x) = 1/cos^2(x) - 1,
            = 0.13247431...,
   tan(x) = 0.36397020...,

or, to the nearest millionth,

   tan(x) = 0.363970.

I believe you can do the above calculations without a calculator,
although having one would be a great labor-saver.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Calculus

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