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Post date: Apr 22, 2012 4:58:30 PM
CIRCE EXPLAINS:
Trigonometric Inverses
Lyn: Shall we start with the diagram from Part 2?
Circe: Not much point. Inverse trigonometry involves turning those straight lengths into central angles - unless you're in radian mode. In fact, we can't even start by naming these inverses, since as I've said, their notation is ridiculous.
Lyn: You mean they don't have names?
Circe: They do - but what's the inverse of Sin?
Lyn: Sin –1 ...?
Circe: That's not a name, that's a notation. Moreover, an "exponent" of -1 usually means creating fractions, but here it means anything but... after all, using denominators was the reciprocal ratios last time!
Lyn: Well... reciprocal does mean multiplicative inverse...
Circe: Stop that! I've got a perfectly good argument here against inverse function notation. By the way, you do know trigonometric inverses are only valid functions for a 180 degree span?
Lyn: Well... technically you can inverse for larger angles, it merely uses their principal angle...
Circe: Stop with the logic! Anyway, you've taken us off topic. The correct answer to the inverse of Sin is Arcsin. Because two points on a circle create an arc. On the diagram, it would be minor arc PC. That's what a calculator returns, radian arc length or degree of central angle.
Lyn: Arcsin? That's not what it says above Sin on a calculator.
Circe: Which is because of stupid oversimplification! This means that, because of the confusion, our inverse of Sine - or ASin - decided to call himself Nis.
Lyn: Oh. So the inverse of Cosine would be Soc... which is really Arccos?
Circe: And Tangent's inverse is Arctan who goes by Nat, yeah.
Lyn: And since they're inverses, you give them a ratio, and they'll return an angle.
Circe: Sure. Or more specifically, they'll give the same angle A we had on that diagram from before - 60 degrees.
Lyn: All right. So we feed Arcsin, Arccos, or Arctan the proper length in red... they'll all give that same answer.
Circe: In fact, if we're in radians (connecting circumference), not degrees, that angle answer is actually arc length PC on my unit circle.
Lyn: So how do they know what value to generate...?
Circe: (sighs) Inverses reflect the primary trig graphs in you? The line y=x?
Lyn: Oh, right.
Circe: Then there's Arcsec, Arccsc and Arccot. Also only definable within 180 degrees, or pi radians.
Lyn: The reciprocal inverses.
Circe: Yeah. Overkill, if you ask me.
Lyn: Since they not only replace X with Y, they're one over the secondary ratios.
Circe: Sure. Means there's a female Csc and a male inverse csC, I mean who needs that?
Lyn: Ah. Well... okay then. (pauses) I don't feel like we learned as much in this segment.
Circe: How about the fact that of the three primary inverses, only Arccos defines himself from 0 to 180, the other two go from -90 to +90?
Lyn: I'm not sure how people can try that at home.
Circe: They can think about why.
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