Do the terms "tangent" (the trigonometric ratio), which is defined as "opposite-side / adjacent-side", and "tangent" (of the curve), which is a line that touches a curve at a point, have the SAME meaning? If yes, how? Also: Is this same for "secant" and "cosecant"?
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J. M. ain't a mathematician
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Sathasivam K
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I think so tangent of a curve is the ratio of change in y-axis(opposite side) to change in x-axis(adjacent side) when the change is very small. – H G Sur Aug 31 '16 at 14:41
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1You might find the following question on the English Language & Usage SE helpful: What reasoning is behind the names of the trigonometric functions “sine”, “secant” and “tangent”? – Michael Seifert Aug 31 '16 at 14:41
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Do You mean $\frac{dy}{dx}$?@H G Sur – Sathasivam K Aug 31 '16 at 14:47
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@Michael Seifert the article that you linked is helpful.but is there any article to know it completely ? – Sathasivam K Aug 31 '16 at 14:54
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2same meaning? No. One's a real function, the other is a geometric object. But related with a very direct application from one to the other. Yes. Definitely. If you take a unit circle and measure the distance from the point of intersection to the point where the tangent line intersects the x axis that value is $\tan$ of the angle of the x axis to center of circle to the point of tangent. Similar for secant and cosecant. – fleablood Sep 01 '16 at 00:16
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The value of angle function found by way of drawing a tangent has been given the same appellation , the same word. – Narasimham Sep 02 '16 at 09:41
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Related: https://math.stackexchange.com/q/1246168/ – DonielF Oct 10 '17 at 18:42
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No, they are not the same.
The tangent function $\tan$ is a function defined on numbers: you put in one number (namely an angle) and obtain another number (namely the quotient between opposite and adjacent leg in a right triangle with the given angle). The property of a line to be tangent to a given curve is something different. As is the operation of finding the tangent to a curve at a given point on the curve. The input here is a curve and one point on it, the output is a line. Since a line is not a number (at least in general), the two things are different.
That doesn't mean the two are not related. The post What reasoning is behind the names of the trigonometric functions “sine”, “secant” and “tangent”? pointed out by Michael Seifert contains a picture which demonstrates how various trigonometric functions can be seen as lengths related to a right triangle of unit length hypothenuse. There the value of the trigonometric tangent function is indeed the length of a certain segment along the tangent of the unit circle around one corner. But being related is not being the same.
