## Monday, October 6, 2014

### New Trig Functions to Learn

In September of 2013 “The Onion” published an article stating that high school students are going to have to learn and master not only the six trig functions (cosine, sine, tangent, secant, cosecant, and cotangent), but also 27 new trigonometric functions.  Fortunately, most of us know that “The Onion” is known for their prank articles.  You can read this article online at:
“Scientific American” was quick to respond to this hoax.  However, they did point out that there are 10 trig functions that we do not teach.  Their names and definitions are listed below:
Versine: versin(θ)=1-cos(θ)
Vercosine: vercosin(θ)=1+cos(θ)
Coversine: coversin(θ)=1-sin(θ)
Covercosine: covercosine(θ)=1+sin(θ)
Haversine: haversin(θ)=versin(θ)/2
Havercosine: havercosin(θ)=vercosin(θ)/2
Hacoversine: hacoversin(θ)=coversin(θ)/2
Hacovercosine: hacovercosin(θ)=covercosin(θ)/2
Exsecant: exsec(θ)=sec(θ)-1
Excosecant: excsc(θ)=csc(θ)-1

This diagram shows the versine, the exsecant, the excosecant on a unit circle.

The trigonometric functions can be constructed geometrically in terms of a unit circle centered at O. This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow.[1] If the arc ADB is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
These functions were used for a few reasons.  Primarily they were used for calculations involving navigation, or to make certain calculations easier or more accurate.  They are not used today because our use of hand held calculators and computers make the use of the use of these functions unnecessary.  You can read the “Scientific American” article at: http://blogs.scientificamerican.com/roots-of-unity/2013/09/12/10-trig-functions-youve-never-heard-of/.
You can also read a more detailed description of these forgotten trig functions at Wikipedia: http://en.wikipedia.org/wiki/Versine and http://en.wikipedia.org/wiki/Exsecant.

David