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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