Library transcendental

The Transcendental library contains functions on numbers.  It includes commonly used numbers, such as π and ℯ, and the transcendental functions, including sine, cosine, and tangent, and their inverse (arc) and hyperbolic counterparts.  Also included are the logarithmic family of functions (the various log functions and exp), and the square-root and power (^) functions.

Sin, cosine, tangent, and their inverse (arc) functions

To understand the transcendental functions (particularly, sine, cosine, and tangent, and their friends), one must know about the relationships for a right triangle.  The discussion will use the below reference figure:

A
|\
| \
|  \
|__θ\
B    C

Sine, cosine, and tangent are used when you know an angle and a length of one of the sides of a right triangle, and you want to know the length of another side.  For these functions, the angle (θ) is in radians, not degrees.  Using the reference diagram above, for sine, you work with the hypotenuse (AC) and the height (AB), for cosine, you work with the hypotenuse (AC) and the base length (BC), and for tangent, you work with the base and the height.

The next question that comes up is if you have two sides of the right triangle, and you need to know the angle.  The arc functions of sine, cosine and tangent resolve this issue.  Let’s use the same reference diagram for the following discussion.

The arc-sine function (asin, also known as the inverse sine) gives θ (the angle in radians) from the height (AB) over the hypotenuse (AC).  The other arc functions, acos and atan, behave as their more-familiar counterparts.  So, the arc-cosine gives θ from the base (BC) and the hypotenuse (AC); the arc-tangent, from the height (AB) and the base (BC).

The π and ℯ constants

So, perhaps you are used to thinking in degrees, not radians.  How do you convert between the two when you use the above functions?  That’s as easy as pi!  If you have the angle in degrees and one of side lengths, first multiple that angle by π / 180.0 (either $double-pi or $single-pi) to get radians, and then use that resulting θ in the appropriate function.

There are several engrossing exercises for computing π, and several novel approaches.  The most widely-known approach is to compute the value of the series 4 - 4/3 + 4/5 - 4/7 ...  (which takes a very large number of terms to get close to Chuck Moore’s find; see $double-pi).  Another is to use the Fibonacci numbers (see the article on computing π using arc-tangents at http://www.mcs.surrey.ac.uk- /Personal- /R.Knott- /Fibonacci- /fibpi.html) which become usable in about ten terms.

I will leave aside the temptation to find mathematical similarities between π and ℯ, with the exception of stating that both can use the Fibonacci numbers to find close approximations of their values.  ℯ finds use in logarithmic and exponential calculations.