| Revision as of 07:30, 8 May 2009 editPenubag (talk | contribs)Extended confirmed users, Pending changes reviewers, Rollbackers8,446 edits add table← Previous edit | Revision as of 16:03, 8 May 2009 edit undoMichael Hardy (talk | contribs)Administrators210,279 edits Deleting nonsense. The trigonometric secant, equal to one over cosine, is NOT what this article is about.Next edit → | ||
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| It can be used to approximate the ] to a ], at some point ''P''. If the secant to a curve is defined by two ]s, ''P'' and ''Q'', with ''P'' fixed and ''Q'' variable, as ''Q'' approaches ''P'' along the curve, the direction of the secant approaches that of the tangent at ''P'', assuming there is just one. As a consequence, one could say that the ] of the secant's ], or direction, is that of the tangent. In calculus, this idea is the basis of the geometric definition of the ]. | It can be used to approximate the ] to a ], at some point ''P''. If the secant to a curve is defined by two ]s, ''P'' and ''Q'', with ''P'' fixed and ''Q'' variable, as ''Q'' approaches ''P'' along the curve, the direction of the secant approaches that of the tangent at ''P'', assuming there is just one. As a consequence, one could say that the ] of the secant's ], or direction, is that of the tangent. In calculus, this idea is the basis of the geometric definition of the ]. | ||
| A ] is the portion of a secant that lies within the curve. | A ] is the portion of a secant that lies within the curve. | ||
| The secant can be calculated as ''1 / cos θ'' | |||
| {{clear}} | |||
| {| class=wikitable align="right" style="margin-left:1em" | |||
| ! style="text-align:left" | '''Function''' | |||
| ! style="text-align:left" | '''Abbreviation''' | |||
| ! style="text-align:left" | '''] (using ])''' | |||
| |- style="background-color:#FFFFFF" | |||
| | ''']''' | |||
| | sin | |||
| | <math>\sin \theta \equiv \cos \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\csc \theta}\,</math> | |||
| |- style="background-color:#FFFFFF" | |||
| | ''']''' | |||
| | cos | |||
| | <math>\cos \theta \equiv \sin \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\sec \theta}\,</math> | |||
| |- style="background-color:#FFFFFF" | |||
| | ''']''' | |||
| | tan<br />(or tg) | |||
| | <math>\tan \theta \equiv \frac{\sin \theta}{\cos \theta} \equiv \cot \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\cot \theta} \,</math> | |||
| |- style="background-color:#FFFFFF" | |||
| | ''']''' | |||
| | csc<br />(or cosec) | |||
| | <math>\csc \theta \equiv \sec \left(\frac{\pi}{2} - \theta \right) \equiv\frac{1}{\sin \theta} \,</math> | |||
| |- style="background-color:#74C0EF" | |||
| | ''']''' | |||
| | sec | |||
| | <math>\sec \theta \equiv \csc \left(\frac{\pi}{2} - \theta \right) \equiv\frac{1}{\cos \theta} \,</math> | |||
| |- style="background-color:#FFFFFF" | |||
| | ''']''' | |||
| | cot<br />(or ctg or ctn) | |||
| | <math>\cot \theta \equiv \frac{\cos \theta}{\sin \theta} \equiv \tan \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\tan \theta} \,</math> | |||
| |} | |||
| ] | ] | ||
Revision as of 16:03, 8 May 2009
A secant line of a curve is a line that (locally) intersects two points on the curve. The word secant comes from the Latin secare, for to cut.
It can be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P, assuming there is just one. As a consequence, one could say that the limit of the secant's slope, or direction, is that of the tangent. In calculus, this idea is the basis of the geometric definition of the derivative. A chord is the portion of a secant that lies within the curve.
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