Trigonometry as the Study of Circles

I recently read John Derbyshire’s book, Unknown Quantity: A Real and Imaginary History of Algebra (Plume 2007 edition). I recommend it overall, although the second half becomes increasingly inaccessible to the layperson. One bit that particular stuck in my head, because of the way it caused me to rethink a mathematical concept, was this passage (p. 87):

Trigonometry–the study of numerical relationships between the arc lengths and chord lengths of a circle–is full of long formulas involving sines, cosines, and their powers.

Two things stood out to me. First, this definition flew in the face of how I would have defined trigonometry; second, it’s completely matter-of-fact, as if Derbyshire doesn’t feel like he’s presenting anything novel here.

As it is taught in high school, the primary object of study in trigonometry is the triangle. This is what’s suggested by the name: Tri- is three, -gono- is corner, -metry is measurement. Trigonometry, the study of the measurement of triangles.

However, Derbyshire’s definition comes to me at a somewhat fortuitous time in that I’ve been noticing lately the relationship between triangles and circles. The Circle Equation (\((x-h)^2+(y-k)^2=r^2\)) is a special case of the Pythagorean Theorem (\(a^2 + b^2 = c^2\)). A circle is the set of points equidistant from a specific point; if we lay out a circle on a coordinate plane, then for all but four of the points on that circle (the intercepts), we can draw a triangle the hypotenuse of which is the same as the radius of the circle, and the sides of which represent the x and y coordinates of the point on the circle.

Also, two major terms that are associated with trigonometry (the tangent and the secant) are also associated with the circle. In the case of a right triangle, the tangent is the measure of the side opposite an angle divided by the non-hypotenuse side adjacent to it, while the secant is the measure of the hypotenuse divided by the non-hypotenuse side adjacent to an angle. In the case of a circle, a tangent is a line that touches the circle exactly once, while a secant is a line that touches it exactly twice.

Tangent

In my student days, I’d written the double use of “tangent” and “secant” off as historical accident. However, teaching geometry this spring, I noticed an intimate relationship between the tangent of a triangle and the tangent of a circle. Consider a circle with a tangent line. Draw a radius from the center of the circle to tangent line. Now draw a second line segment from the center of the circle to any other point on the tangent line. You’ve just drawn a right triangle. The tangent of the angle with its vertex in the center of the circle is the length of the line segment on the tangent of the circle divided by the radius of the circle (\(\tan(\angle DAB) = \overline{BD}/\overline{AB}\)). The word “tangent” itself comes from Latin “tangere”, “to touch”; the non-mathematical sense of “tangent” implies having a shared conceptual point but going off in another direction. Linguistically, it seems that the first mathematical relevance of “tangent” was in regards to measuring circles and the second, later sense was in measuring triangles.

There’s a similar pattern with the secant, although it’s a bit more complicated. Draw a line secant to a circle. Draw the radii connecting the center of the circle to the two points of intersection. Finally, draw a line segment perpendicular to the secant line going to the center of the circle. You now have two right triangles. For each triangle, the ratio of the radius of the circle to the part on the secant line represents the secant of the angle on the circle (e.g., \(\sec(\angle ABD) = \overline{BA}/\overline{BE}\)); the length of the entire chord can be determined by adding the secants of the two angles. The word “secant” comes from Latin “secare”, “to cut”; again, this suggests that the circle-measurement use came first.

Putting this together, referring to trigonometry as the study of triangles answers the what, but not as succinctly the why. Certainly triangles are an interesting mathematical object in their own right; we can use trigonometric functions to measure the height of tall buildings, for instance, based on either the shadows the cast or how high we need to look up to see the top edge. However, from a historical perspective, the circle has been a far more elusive object than the humble triangle. Consider the amount of mathematical time spent computing π alone, the initial prime use of which being to determine the circumference a circle (π, meanwhile, has since popped up in a variety of mathematical places, many having little or nothing to do with circles).

So while high school trigonometry is taught largely in terms of triangles, it seems that Derbyshire’s why of trigonometry being tied to circles has a strong argument, at least from a historical perspective. (Illustrations created in GeoGebra 4.)

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