What Is a Secant?
Question
When cosine is zero, secant stretches to
Hi, I'm Jen D'Amore for About.com and this video is all about secants. What is a secant?

Definition of a Secant

"Secant" comes from the latin secare, which means to cut. In geometry a secant line is a line that cuts, or intersects, like this secant that is intersecting the curve. You may see a secant slicing through a circle or an open curve, in geometry, or on a graph.

Chords Are Segments of a Secant

The segment of the secant that is on the inside of the curve is called a chord. And if the two points that intersect the curve become close enough together, the secant becomes a tangent.A secant line on a function joins two points and is the slope, or the average rate of change.

Secants in Trigonometry

In trigonometry, the secant of an angle in a right triangle is the ratio of the hypotenuse to the adjacent leg, so it's the hypotenuse divided by the adjacent leg. Secant is abbreviated as sec and is the reciprocal of cosine.

Secant Is the Reciprocal of Cosine

So if sec is not on your calculator, you can use 1/cos x to replace sec x. sec x = 1 / cos x. If you compare the graph of cosine and secant, you can see that they are opposite. When cosine is zero, secant stretches to infinity.

Intersecting Secants Theorem

There are of course a few theorems utilizing secant, including the intersecting secants theorem, which states that if two secants pass through a circle and intersect, the products of their segments will be equal. In this diagram we have two secants intersecting outside the circle. So PA x PB = PC x PD regardless of where they fall on the circle.

Tangent-Secant Theorem

The tangent-secant theorem: Similarly, if the two points of one of the secants are overlapping, that secant becomes a tangent, so PA would equal PB. Therefore, the formula would alter to PA^2 = PC X PD which is the tangent-secant theorem. Not far from that, if the second line was also a tangent, PA^2 = PC^2.

Secant Method Is a Variation on Newton's Method

Looking back at our function, the secant method is a variation on Newton's method, and is a way to find, or approximate, the root of a function by drawing two secant lines through the function and using the intersections to plug into the equation.