Wednesday, June 11, 2008
Wednesday Math, Vol. 27: The trig functions
One year and one day ago, I wrote a post about the trig functions. Back then, I think the only people reading my stuff were just a few blog buddies and some family members, though I did get flags from Poland and South Korea that day. Still, it's not exactly repeating myself to do another post about trigonometry, which is an important part of mathematics, especially since I have a new and hopefully less confusing picture. (Click on the picture for a bigger version.)
Translating the roots of the word, trigonometry means "triangle measure", and if we have an angle, it will define a set of similar right triangles when combined with the horizontal axis. We are interested in three of the similar triangles, the ones where one side of the triangle has measure equal to 1.
When the hypotenuse is equal to 1, the horizontal side length is equal to cosine and the vertical side is equal to sine. The prefix "co" is short for complementary, which in geometry means two angles that add up to 90 degrees. So, for example, since 37 + 53 = 90, the sine of a 37 degree angle is the complementary sine (cosine) of a 53 degree angle, and vice versa.
In the second picture, the side of length 1 is the horizontal edge, and the other two line segment lengths are tangent and secant. In trig, students are usually taught that tangent = sine/cosine and secant = 1/cosine, which in some ways make it trickier to remember, because it would be easier if secant was related to sine and cosecant related to cosine, but they aren't. Tangent goes by another name in coordinate geometry. It is rise/run, which means it's the slope of the line.
In the third picture, the vertical edge is of length 1, the hypotenuse is cosecant and the horizontal is cotangent. The major trig identities, each one listed under the corresponding picture, are all versions of the Pythagorean Theorem, since it's all about right triangles.
It's very old school, and I mean before-the-birth-of-Christ old school, to measure angles by degrees, and in trig classes the new way, and I mean new as in Newton, to measure is in radians. All the way around the unit circle is 2π instead of 360 degrees, and radians are often given as multiples of π. For example, 45 degrees is 1/8 of the circle so 2π*1/8 = π/4.
So that's what you should know after your first hour of a trig class, which means that's what you should remember 20 years after you take a trig class, if you're lucky.
Hope you were taking notes.