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Post date: May 27, 2012 7:1:28 AM
CIRCE EXPLAINS:
Older Trig
Lyn: Okay, so what's a "versine"? And why are we fighting her?
Circe: It's versed sine. 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. Before calculators.
Lyn: O-kay... so if I wanted to multiply 20 times 30...
Logan: You would do log 20 plus log 30, which is about 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? Like for log 20?
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 too. 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 trig losing another 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 exterior to my circle, 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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