# What good is the secant?

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Students no sooner learn about the sine and cosine functions, which are strange enough at first, when they are exposed to . Years later, they may reflect (as I have) that they never used the  function directly. What were mathematicians thinking? Well, they may have been thinking about cartography. The secant can be used to compute projections of points from the sphere onto a map.

In any case, each of the trig functions has a simple geometric interpretation that makes it clear the names are not arbitrary. Consider a unit circle (radius is 1), and an angle  that is measured from a horizontal ray. The other ray of the the angle intersects the circle. The vertical distance is sine, and the horizontal distance covered is the cosine (the sine of the complementary angle). So far, so good.

Now, draw a line segment tangent to the circle on the bottom ray of the angle. Where the other ray intersects the tangent, the vertical distance covered is the tangent function.

The secant line segment cuts the circle and intersects the tangent line. The distance from the origin to the intersection is the secant function. Note the latin root of secant is secans, "to cut."

Finally, we have the cofunctions cosecant and cotangent. In the same way that cosine is the sine of the complementary angle, they are simply the secant and tangent of the complementary angle.

It is curious that the typical math class introduces sine and cosine first. What if we started with sine and secant? Geometrically, can the student see what happens if we multiply the sine by the secant? I feel like we should spend more time with unit circles, straightedges, and geometry to create these functions and discover various trig identities. We dive too quickly into the function abstraction.

In the sketch below, what do the colored lines mean? Where would you draw the corresponding lines for the co-functions?