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The definition of the slope of the line of the secant is:

slope = $\frac{y2-y1}{x2-x1}$

The definition of the slope of the tangent line is:

$\lim_{h->0}\frac{f(x+h)-f(x)}{h}$

I understand why they call it the tangent line since the angle to the x axis will be $tan(\theta) =\frac{Opp}{Adj}$ equivalent to opposite of adjacent.

Secant is the inverse trig function of cosine, so $\sec(\theta)=\frac{Hyp}{Adj}$

But I don't understand how secant is related to the slope of its line? I looked it up and I found out that the word secant comes from the Latin word secare, which means to cut. But is there any relation to secant and it's angle?

Klik
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  • I don't think tangent line is called that because of $\tan\theta = opp/adj$, I think they both derive their meaning separately from whatever common root word tangent comes from. – nullUser Jun 06 '13 at 01:30
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    Perhaps. This was also a thought I had, but I'd like to think there is another reasons since, tangent and secant both have specific definitions associated with them. I thought it would be unlikely that mathematics gave them specific definitions in one circumstance and then a more flexible definition in another. But then again, what do I know. – Klik Jun 06 '13 at 01:42
  • The trigonometric functions $\tan$ and $\sec$ have no real relation to tangent lines or secant lines. –  Jun 06 '13 at 01:43
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    @Lenny That is false. See the picture here. – Pedro Jun 06 '13 at 01:48
  • /facepalm/ Didn't read the last paragraph. My apologies. – Cameron Buie Jun 06 '13 at 02:01

1 Answers1

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Because you can define $sec(\theta)$ as a length on the unit circle. $sec$ corresponds to the length of the line from $(0,0)$ to $(1, \tan(\theta))$ and $tan$ corresponds to the length of the segment from $(1,0)$ to $(1, \tan(\theta))$. See the figure here. Clearly the $sec$ segment cuts the circle and $tan$ is tangent to it.

John Strood
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Tpofofn
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