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10 Secret Trig Functions Your Math Teachers Never Taught You

On Monday, the Onion reported that the "Nation's math teachers introduce 27 new trig functions
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The gamsin, negtan, and cosvnx from the Onion article are fictional, but the piece has a
kernel of truth: there are 10 secret trig functions you've never heard of, and they have delightful names
like "haversine" and "exsecant
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(It's well known that
you can shake a stick at a maximum of 8 trig functions
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The versine is in green next to the cosine, and the exsecant is in pink to
the right of the versine
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Not pictured: vercosine,
covercosine, and haver-anything
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Johnson, via Wikimedia Commons
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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

I must admit I was a bit disappointed when I looked these up
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Why did they even get names?! From a time and place where I can sit on my
couch and find the sine of any angle correct to 100 decimal places nearly instantaneously using an
online calculator, the versine is unnecessary
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Numberphile recently posted a video about Log Tables, which explains how people used logarithms to

multiply big numbers in the dark pre-calculator days
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The equation
logbx=y means that by=x
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One handy fact about logarithms is that
logb(c×d)=logbc+logbd
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If you wanted to
multiply two numbers together using a log table, you would look up the logarithm of both numbers and
then add the logarithms together
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It sounds cumbersome now, but doing multiplication by hand
requires a lot more operations than addition does
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The secret trig functions, like logarithms, made computations easier
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Near the angle θ=0, cos(θ) is very close to 1
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To illustrate, the cosine of 5 degrees is 0
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999847695
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003652997
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And a table with only three significant figures of precision would not be able to distinguish
between 0 degree and 1 degree angles
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The bonus trig functions also have the advantage that they are never negative
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(It is not defined for 0 either, but that is an easy
case to deal with
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A little bit of trigonometric wizardry (a
...
a
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So the haversine
is just sin2(θ/2)
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If you have a computation involving the square
of sine or cosine, you can use a haversine or havercosine table and not have to square or take square
roots
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Image: Qef and Steven G
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The versine is a fairly obvious trig function to define and seems to have been used as far back as 400 CE
in India
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The haversine formula is a very accurate way of computing distances between two points on
the surface of a sphere using the latitude and longitude of the two points
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(On the other hand, the haversine formula does not do a very good job
with angles that are close to 90 degrees, but the spherical law of cosines handles those well
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As recently as 1984, the amateur astronomy magazine Sky &
Telescope was singing the praises of the haversine formula, which is not only useful for terrestrial
navigation but also for celestial calculations
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Math article
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All of them could
make computations more accurate near certain angles, but I don't know which ones were commonly
used and which ones were named* analogously to other functions but rarely actually used
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When the Onion imitates real life, it's usually tragic
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We're very lucky now that we can multiply, square, and take
square roots so easily, and our calculators can store precise information about the sines, cosines, and
tangents of angles, but before we could do that, we figured out a work-around in the form of a
ridiculous number of trig functions
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These functions actually
made computations quicker and less error-prone
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But I think we can all agree that it should come back, if only for the
"awesome" joke I came up with as I was falling asleep last night: Haversine? I don't even know 'er!

*I'd like to take a little digression to the world of mathematical prefixes here, but it might not be for
everyone
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In the table of secret trig functions, "ha" clearly means half; the value of haversine is half of the value of
versine, for example
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(Complementary angles add up to 90 degrees
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) For instance, the cosine of an angle is also the sine of the complementary angle
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The one bonus trig function that confuses me a little bit is the vercosine
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Instead,
the vercosine is the versine of the supplementary angle (supplementary angles add up to 180 degrees),
not the complementary one
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In the case of versine, I
believe the definition involving cos(θ) is older than the definition involving sine squared
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If you're a trigonometry history buff and you have more information, please let me know! In
any case, the table of super-secret bonus trig functions is a fun exercise in figuring out what prefixes
mean

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