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017-Circe Explains Trig, Part 3

posted Apr 22, 2012, 9:58 AM by Gregory Taylor   [ updated Apr 22, 2012, 10:12 AM ]


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. 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.

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, the 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, 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.

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.