Calculus How To

Secant Line: Definition, Examples, Finding

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Contents (Click to skip to that section):

  1. Secant Line Definition
  2. Exsecant Function

1. Secant Line Definition

A secant line (from the Latin Secare, to cut) connects two ore more points on a curve.

The following image shows a secant line that connects two points, along with a tangent line (which skims the curve at one point):
secant line

A secant line is useful to calculate the slope of a line.  A secant line is the equivalent of the average rate of change or the slope between two points.  Once you have calculated the slope of a line we can find the equation of the line through the two points.  The two points of a secant line are denoted by:

(x1, y1) and (x2, y2)

Slope of a Secant Line

The slope of the secant line is calculated using the formula:

(y2 – y1) / (x2 – x1)

The equation of the line through the two points can be found by using the slope-point formula:

y – y1 = m(x – x1)

Finding the Equation of a Secant Line

Finding the equation of a secant line is a three-step process:


  1. Locate two points on the secant line.
  2. Find the slope of the line that runs between the two points.
  3. Find the equation using the point slope formula.

Example:

Question: Find the equation of the secant line to the curve
f(x) = 4x2 – 7 where x = -2 and x = 1.

1. Find Two Points

The question gives us the values for x so we must determine the y values in order to calculate the slope of the line.  We will calculate for x = -2 first:

  • f(-2) = 4(-2)2 – 7
  • = 4(4) – 7
  • = 9

So the coordinates of the first point are (-2, 9).  Next, find the coordinates of the second point x = 1.

  • f(1) = 4(1)2 – 7
  • = 4 – 7
  • = -3

The coordinates for the second

point are (1, -3).

2. Find the Slope

Now that we have the coordinates for the two points we can find the slope of the line.

  • m = 9 – (-3) / -2 -1 = 12/-3 = -4

The slope of the line between the two points is -4. 

3. Find the Equation

Plug the slope into the slope-line formula to find the equation of the line.

  • y – y1 = m(x – x1) = y – (-3) = -4(x – 1) = y + 3 = -4x + 4
  • y = -4x + 1

The equation is y = -4x + 1.

Exsecant Function

exsecant function

The external secant AD (orange) gives the part of the secant segment AC that’s outside the unit circle.



The exsecant function is an archaic function defined as:

exsec(x) = sec(x) – 1 = tan(x) · tan(½ x)

or, defined in terms of an angle (θ):

exsec θ = sec θ – 1

The word is a contraction of external secant.

Meaning of the Exsecant Function

The exsecant function gives you the part of the secant line that’s outside the unit circle, which is a circle with a radius of 1. The segment inside the circle is 1, so all you have to do to find the exsecant is to:

  1. Find the secant of the angle (length AC in the above image),
  2. Subtract 1.

An Archaic Function

The function was once an important part of surveying, astronomy and sea travel, appearing in many nautical and astronomical tables. It is now used infrequently, because tables have fallen out of everyday use and you can calculate the exsecant directly.

References

Chapter 8 Families of Functions. Retrieved December 19, 2019 from: http://www.ms.uky.edu/~droyster/courses/fall06/PDFs/Chapter08.pdf
Trigonometry. Retrieved Decmber 19, 2019 from: https://mysite.du.edu/~jcalvert/math/trig.htm
Oldham, K. et al. (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. Springer Science & Business Media.
Schwatzman, S. (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA.