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016-Circe Explains Trig, Part 2

posted Apr 4, 2012, 4:26 AM by Gregory Taylor   [ updated Apr 4, 2012, 4:31 AM ]


Secondary Trigonometric Ratios

Circe: We'll start with the diagram we already had from Part 1.

Lyn: That's handy. So, the secondary ratios are the reciprocals.

Circe: Precisely. Still lengths. Not the inverses, which are angles.

Lyn: In fact, a secant is actually a line that cuts through a curve. So how is it also a trig function?

Circe: Consider that the reciprocals must always be more than one, being defined as one over their primaries - that is, one over a fraction. So by starting at the origin, secant must cross the circle.

Lyn: Where we define secant as hypotenuse over adjacent.

Circe: But remember our triangle with adjacent of one!

Lyn: So "sec A" is the hypotenuse! The horizontal distance from the origin to the intersection with tangent.

Circe: There it is. Which connects it to the latin root word, and explains why secant is undefined at the same time as tangent.

Lyn: And as the point moves around, I suppose secant cuts through the other side of the circle instead. But now... to do the same trick with cosecant, you'd need an opposite of one.

Circe: Right. (smiles) The tangent line has two axis intercepts.

Lyn: How does that... oooh, we extend that line back to the y-axis. Another right triangle! Is it similar?

Circe: Obviously. Consider the huge 'SAT' triangle now, formed by the tangent line and the axes. A right angle at the origin, and an angle shared with the smaller one we have now.

Lyn: Two equal angles implies three equal angles... or yeah, for that matter, a Z pattern, along with the radius of the circle being the altitude of the big triangle. Okay.

Circe: So angle A is the same as angle S up here at the top. The radius is now opposite to it, therefore the opposite has length one.

Lyn: Therefore cosecant, or hypotenuse over opposite...

Circe: Is merely the hypotenuse of this third triangle.

Lyn: Which defines "csc A" as the vertical distance from the origin to the intersection with the tangent!

Circe: Intersection with the what?

Lyn: The tangent line.

Circe: The line, yes. A rather important distinction, because where is cotangent?

Lyn: It's... oh! Oh, part of the tangent line! The bit from point P to the y-axis. So of course cotangent is zero at 90 degrees! That's also why tan and cot are equal at 45 degrees!

Circe: Precisely! Cotangent is adjacent over opposite! Moreover, we now see COsecant meets COtangent in the same way secant meets tangent.

Lyn: Then "cot A" was along our tangent line all this time... merely the distance going the other way.

Circe: Now you've got it. If you don't believe me, try measurements at home!