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018-Circe Explains Trig, Part 4

posted May 27, 2012, 12:01 AM by Gregory Taylor

CIRCE EXPLAINS:

Older Trig



Lyn: Okay, so what's a "versine"?

Circe: Let me get that diagram back from Part 2...

Lyn: Versine's a length then.

Circe: She is, yes... here. So, the versed sine equals "1 - cos(x)". You can see it right here on the diagram, it runs from the end of cos to the unit circle.

Lyn: What was she used for?

Circe: (sighs) Where to start? I suppose there's a connection to logarithms...

Logan: Yarrrr! About time someone called for me!

Circe: What the...?! Nobody called for you!!!

Logan: One of the early uses for logarithms was to turn multiplication into addition. Most people found addition easier.

Lyn: O-kay... so if I wanted to multiply 20 times 30...

Logan: You would do log 20 plus log 30, which is 2.77815. Since we're in base ten, you then take ten to that power to get your answer, seeing as 10^2.77815 is 600.

Circe: This is Circe Explains, not Logan Explains!

Lyn: That means there would have to be a table, to look up the values?

Logan: Wellll, and log 6 plus two is actually the same decimal, once you know the logarithm rules...

Circe: GO. AWAY.

Logan: Yarrrr....

Circe: Good. But yes, the versed sine had - has? - the advantage of always being positive, meaning you could always calculate a logarithm for it. 

Lyn: Meaning you could use logarithmic tables when multiplying this sort of trigonometry.

Circe: Precisely. Versine is also equal to 2*(sin(x/2))^2 which is a nice little trig identity. Means you can use the versine instead of squaring numbers. Versine is also more precise for small angles.

Lyn: Is... is there a coversine?

Circe: Of course, it's "1 - sin(x)", right here on our diagram. Be careful you don't confuse it with the vercosine, which is "1 + cos(x)".

Lyn: Wait, what?

Circe: The vercosine, it covers the rest of the diameter of the circle. Well, horizontally. As does the covercosine, covering off the diameter vertically.

Lyn: The coverco... is there going to be a test on this?

Circe: Then there's the haversine.

Lyn: Dare I ask?

Circe: Half the versine.

Lyn: Of course it is.

Circe: The haversine formula is still useful today in dealing with spheres. The earth isn't flat, you know.

Lyn: Yet I'm guessing that means there's a havercosine.

Circe: And a hacoversine, and a hacovercosine. Yet despite ALL of that... their graphs? Just transformed sine waves.

Lyn: I'm starting to understand why you're not sad about losing another trig ratio.

Circe: We're not even done. There's also the exsecant.

Lyn: (facepalms) Secant is no more! She has ceased to be! She rests in peace, is pushing up daisies, has kicked the bucket, this is an ex-secant!

Circe: Look, if you're not going to be SERIOUS...

Lyn: Sorry, sorry. I'm guessing exsecant is "sec(x) - 1", the missing part of secant on our diagram here?

Circe: Very good - exsecant is secant minus cosine and versine. It's also the versine divided by cosine.

Lyn: Which implies an excosecant up here. Is there an excotangent?

Circe: Why would you want to subtract 1 from cotangent? Don't you already have enough to keep track of?

Lyn: Yes. I'm not sure people can try any of this at home either.

Circe: Maybe they can ask their ancestors about it.

Lyn: I need to lie down. Find me a cot.

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