Natural Blogarithms

Secants and Tangents

September 7th, 2006 by SplineGuy

I was posed with the question today of how secant lines are related to the secant function. Equivalently, it lead to the question of how tangent lines are related to the tangent function. I vaguely recalled seeing them illustrated with the unit circle but was unable to recall the exact relationship. After a minimal amount of searching, I came across this image in the public domain that helps to explain it.

You’ll notice from the picture that when you consider some angle, $\theta$ , you can draw the secant line passing through the point $(\cos \theta, \sin \theta)$ on the outside of the unit circle (which is the point $A$ in the picture) and through the origin. Now, it is not on the picture but you can then draw the line $x=1$ and the distance from the origin to where the secant line intersects $x=1$ is the value of the $\sec \theta$ . This is equivalent to the length of the line $OE$ in the image. You can also see from the image, how the tangent line at $A$ is related to the $\tan \theta$

You may be wondering, what in the world are versine and exsecant? Well, I didn’t know either, but it turns out that they were common historically (and appeared in the earliest tables), but are now seldom used:

$\mathrm{versin} \theta = 1 - \cos \theta$