I was playing with a graphing calculator program for my PDA the other day and I made A Great Mathematical Discovery. I figure that like most of my Great Mathematical Discoveries, it was stolen from me 2500 years or so before my birth by the Greeks. I’m used to this.
Anyways, I discovered that–
(sin(x))^2+(cos(x))^2=1
It seems to me that there is a reason for this and I tried to think back to Algerbra II and Trigonometry class from high school (this was when Richard Nixon was President, by the way) and the only thing I can remember from the class was that a varsity cheerleader named Denise Nemeth, who did not dress conservatively, sat in front of me and despite the fact (or perhaps because of the fact) that I was the geeky sophomore who got straight A’s in a class full of juniors and seniors, she would actually talk to me. In my entire high school career, she was the only cheerleader to talk to me.
At any rate, I can’t remember any of the actual math. Is there a reason why (sin(x))^2+(cos(x))^2=1?
It’s just the Pythagorean Theorem in disguise. Remember that a^2 + b^2 = c^2, where a and b are the two short legs of a right triangle, and c is the long leg (the hypoteneuse), opposite the right angle? well, divide both sides of that equation by c^2, and remember that a/c = sin (theta) and b/c = cos (theta), and the rest follows.
If you don’t recall the Pythagorean Theorem, or its many proofs, use any search engine. Someone once published a book entitled "100 Proofs of the Pythagorean Theorem’, and they’re amazing – you can do it many ways without repeating yourself.
And in addition to the fine answers given by CalMeacham and Cunctator, I found that pretty much all of Trig, Analytical Geometry and a least 1/2 of Calculus (the easy half), is based on that equation in one of it’s many forms. I’m too many years removed from my math classes to give any gee whiz examples, but they’re out there.
The functions sine and cosine are defined using a unit circle. Let’s use Q for our angle, because x is too suggestive. So what does sin Q mean? If you take a unit circle (that is, with radius 1) whose center is at the origin, and you draw a radius which makes an angle Q with the x-axis, then sin Q is the y-coordinate of the point where your radius meets the circle. Likewise, cos Q is the x-coordinate of that point.
It is here that radians become a more natural unit than degrees. If you start at the point (1,0) on your circle, that is, where the circle meets the x-axis on the right, and you start dragging your pencil around the circle counter-clockwise, then the distance your pencil travels (as the pencil draws, not as the crow flies) is the number of radians. So if you draw all the way around the circle, you have moved your pencil 2pi units and that corresponds to 2pi radians. So radians are secretly a measure of distance.*
Again, if you consider an angle of Q radians, the sine of that angle is the distance above the x-axis your pencil point is after you’ve travelled Q radians around the outside of the circle. The reason that sin^2 + cos^2 = 1 is that you’re taking the two sides of a right triangle made from a radius of your circle and two sides parallel to the x- and y-axes.
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*radians should actually be thought of as a measure of distance on a circle where you express the units in terms of the radius of the circle. So if the circle has radius 1 inch, the circumference has length 2pi inches. This is 2pi times the radius, or 2pi radians. If you have a circle with radius 3 feet, its circumference is 6pi feet long, and again this is 2pi times the radius so 2pi radians.
Here’s a diagram of the triangle that I’m talking about above.
Also, a bit of historical information: the earliest application of sines was related to the lengths of chords. A chord of a circle is a segment that starts and ends on the circle. That is, take two points on a circle and connect them together with a line. That line is the chord. Now, how long is the chord? If you cut off an arc along the circle which is Q radians long, then the chord you cut it off with will be 2sin(Q/2) radians long.
As has been said it’s just the Pythagorean Theorem in action. Draw a right triangle with a hypotenuse equal to 1 unit and label one of the acute angles, angle x. The side opposite x is then numerically equal to sin x and the side adjacent is numerically equal to cos x. By the theorem the sum of the squares of those two sides is equal to 1[sup]2[/sup].
This (and indeed, all of trigonometry) comes from the most useful equation in all of mathematics: I give you Euler’s Formula. This relates complex analysis (i.e. mathematics using both the real and imaginary axes) to circular or periodic motion, and finds use in everything from plotting parametric circular or eliptical functions in terms of an angular variable (i.e. plane trigonometry) to vector analysis and signals processing. It’s the basis for Fourier analysis (generating a composite function out of sine and cosine functions).
You can, if so inclined, derive all trigonometric functions, from the identity discovered (if somewhat belatedly) by the OP, to the half-angle formulas. It also gives us the most beautiful equation in all of natural mathematics, Euler’s Identity, which encasulates the five most important numbers of all time.
So, not a coincicence; it’s fundamantal to trignonometry and analytical geometry. Now, let me introduce you to the Hardy-Ramanujan number, 1729.
