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[Heron TSG]

So I've been doing pre-calculus, and we're on Trigonometric Identities. Secant(X), one of those, is equal to 1/Cos(X).

There is also a geometric term which consists of a line segment connecting any two points on a circle. Are these related?

More questions may come later.

[G-Flex]

A line segment whose endpoints exist along the same circle is a chord. Not sure what relationship they have to the secant function, though.

[Heron TSG]

Well, it turns out that a secant is the entire line, not just the segment. (which is what a chord is.)

I've been told they are related via deep magic, but nothing is turning up.

[Karlito]

A line (not a segment) going through two points on a circle (or any curve) is called a secant line. I believe there is a relationship between a secant line on the Unit Circle, and the secant function, just as there is a relationship between a tangent line on the Unit Circle and the tangent function, however, I can't adequately explain either of those relationships without looking it up and you don't need to understand them to do the work.

[Vector]

So I've been doing pre-calculus, and we're on Trigonometric Identities. Secant(X), one of those, is equal to 1/Cos(X).

There is also a geometric term which consists of a line segment connecting any two points on a circle. Are these related?

More questions may come later.

Heehee, I think I like you.  A secant line is one connecting two points of a curve.  When you move those two points closer and closer together, you get a tangent line (the straight line which most fully approximates the curve at a given point).  I don't think the names are related to the trigonometric functions, however [in relation to a non-circular curve].  Well... they sort of are, but in a convoluted way.  It has more to do with a special way of finding tangent lines, IIRC, but we usually use calculus nowadays.

I can't adequately explain either of those relationships without looking it up and you don't need to understand them to do the work.

^ This is why we don't get to have great mathematicians anymore.

[Karlito]

There's great mathematicians, you just have to have a doctorate in modular functions or somesuch to get to level where discoveries can be made. No one cares about Euclidean geometry any more.

Anyway, after reading wikipedia, I understand the tangent function fairly well, though I'm still iffy on the secant.

Quoth Wikipedia:

Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (as shown in the picture to the right), and similar such geometric definitions were used historically.

    * In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India[1] (see history).
    * cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD.
    * tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF.
    * sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively.
    * DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle).
    * From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.[2])

Unhelpful Picture:

[Vector]

No one cares about Euclidean geometry any more.

Perfidy.

[Heron TSG]

Does the secant's (the line) slope equal (The value) Secant(X)?

If so, it still doesn't denote where the line goes. I really hope these two things are related and don't just have accidentally similar names.

[G-Flex]

Does the secant's (the line) slope equal (The value) Secant(X)?

A look at the diagram shows this to be false. θ is obviously between 0 and 90 degrees, meaning that sec(θ) can't be zero, but the slope of the line is zero.

[Heron TSG]

So, what if I were to propose that the secant(X) was the percentage of the circle's area situated to the left (or above, if horizontal) of the secant line?

[G-Flex]

Think about that for a second, though.

sec(x) = 1/cos(x)

The range of cos(x) is between -1 and 1. This includes the area around and including zero.

sec(x) is the reciprocal of that, so it gets very very high, actually ranging between negative and positive infinity.

Obviously, you're not going to have, say, 11679857% of a circle's area to the left of any line, no matter where it is.

[Heron TSG]

what if you inverse that number though, and then take it as a fraction?

instead of 11679857%, you'd get 8.56x10^8, which makes sense. It's not zero, but it gets close. You'll never have a secant with NO length, so you never will reach zero.

[Vector]

what if you inverse that number though, and then take it as a fraction?

instead of 11679857%, you'd get 8.56x10^8, which makes sense. It's not zero, but it gets close. You'll never have a secant with NO length, so you never will reach zero.

... You are not speaking English (or, rather, you are not speaking mathematics).  First off, 8.56x10^8 is a rather large number.  Second off, a secant with no length is called a point.  Third, I think you're talking about the cosine.

[G-Flex]

Yeah, the "inverse" (I assume he means reciprocal) of the secant is just the reciprocal of the reciprocal of the cosine... in other words, the cosine.


I'm assuming he meant8.56x10^-8, too. That's the only way it makes sense, at least.


Granted, it's wrong anyway. Draw a secant line pointing more-or-less vertically. The angle is then 90 degrees. The cosine of 90 degrees is zero. Obviously, more than none of the area of the circle is to the left of it.

[Heron TSG]

First, 8.56x10^8 should be 8.56x10^-8. whoops.

Second, the point (derp.) is that you won't have a secant when it becomes a point. you'll have a point. Therefore, the relationship would still work because it'd be true for all secants.

Third, An inverse relationship is still a relationship. It may not be a function, however. That;d explain the wackiness dividing by zero would entail.

Pseudo-edit: I know I'm probably wrong, I'm just chucking ideas out in hopes of a bullseye. It'd be nice to learn something without extensive help from textbooks and teachers.