Rational points on the unit circle and the Pythagorean triples.

Trigonometry translated from it Greek roots means "the measure of triangles", most notably right triangles. The most important aspect of right triangles is the Pythagorean Theorem, usually stated as a² + b² = c².

The unit circle on the xy plane is given by the formula  x² + y² = 1. There are infinitely many points on the circle and if you pick one truly at random, it will very likely be irrational, which means neither the x or the y be written as p/q, where p and q are whole numbers. The only way it is possible for a point on the unit circle to be rational is to have two fractions of the form a/c and b/c where a² + b² = c². For example,

(3/5)² + (4/5)² = (5/5)² = 1.

In Excel, I generated 26 rational points on the unit circle, the largest denominator being 1,105. As you can see, there are a lot of gaps, but you can make out the general shape of a quarter circle.

Here's what it looks like with 662 rational points plotted. You can still see some gaps, notably near the horizontal lines at 0.8 and 0.6. If all the rational points were plotted, and there are infinitely many, you would not be able to see any gaps with the naked eye or at any magnification. In math, we say that the set of rational points is dense on the unit circle.

But here's one of the goofy things about infinity. There are also infinitely many points on the unit circle where both values are irrational AND infinitely many points where x is rational and y irrational AND infinitely many points where x is irrational and y rational. Let me write those sets out.


Set #1: { the set of points on the unit circle where both x and y are irrational}
Set #2: { the set of points on the unit circle where both x and y are rational}
Set #3: { the set of points on the unit circle where x is rational and y irrational}
Set #4: { the set of points on the unit circle where x is irrational and y rational}

All four of those sets are dense on the circle, meaning that in any segment of the circle no matter how small, there will be elements from all four of the sets.

Here's something even weirder about infinity. Sets #2, #3 and #4 are all the same size of infinity, but Set #1 is a bigger size of infinity, bigger than the other three combined.

Tomorrow, we move from goofy facts about infinity to more practical math aimed at answering the question "Is climate change or global warming real?"

The Pythagorean Theorem and its connection to trigonometry.

When teaching math, if I ask a class "What is the Pythagorean Theorem?", invariably someone will eventually pipe up the statement

a² + b² = c²

This is correct as far as it goes, but asking what a, b and c stand for is not always remembered.

Ignoring what names we assign the sides the statement of The Pythagorean Theorem is:

The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.

Quite often, the phrase "of the length(s)" is removed, since talking about squaring something implies the thing is a number and the number associated with a side of a triangle is the length almost always.

Using the names from the picture above, the equation would change to adj² + opp² = hyp².

So far, so good.

Let's divide all the sides by the hypotenuse.

adj/hyp = cosine
opp/hyp = sine
hyp/hyp = 1

These are still sides of a right triangle with 1 being the hypotenuse, so the equation changes to a form the is the main trigonometric identity sin² + cos² = 1. When I teach the class, I call this The Trigonometric Identity since all the other trig identities are just re-arrangements of this.

Tomorrow, I'll present a proof of the Pythagorean Theorem using basic facts from geometry, the areas of right triangles and the areas of squares.

The basic trigonometric functions.

If I give you the measure of two interior angles of a triangle, you can figure out the third by adding up the two given numbers and subtracting that number from 180.

In the picture here, the angle on the left is theta degrees and the angle on the lower right is 90 degrees. 180 - (90 + theta) = 90 - theta, which is the measure of the upper right angle.

The two smaller angles must add up to 90 degrees. When two angles have this property, we say the are complementary.  Notice, this is different from complimentary, which means saying something nice. Here are the ways to use complementary in a sentence.

30° and 60° are a pair of complementary angles.

30° is complementary to 60° and vice versa.

That said, the picture above is an example of a class of similar triangles defined by the three angles, 90°, theta° and (90-theta)°. Just as we have not put a number value to theta, we have not put number values to the lengths of the sides, labeled hyp, opp and adj, respective abbreviations for hypotenuse, opposite and adjacent. We could measure theta exactly and give the measure in degrees, but the numbers for the side lengths would depend on what unit we used. Americans would be likely to measure in inches, most of the rest of the world would use centimeters or millimeters, and some clever computer types might try to figure out the number of pixels in each line as drawn in the computer picture.

If the measurements are exact, the numbers produced will be different but the ratios of matching side lengths should be the same, like the hypotenuse divided by the adjacent or the opposite divided by the hypotenuse.  Since we have three numbers, none of which should be zero, we have six different ratios. In trigonometry, there are six basic functions we will associate to an angle measurement, in this case theta.

sine: sin theta = opp/hyp

cosine: cos theta = adj/hyp
tangent: tan theta = opp/adj
cotangent: tan theta = adj/opp (the reciprocal of tangent)
secant: sec theta = hyp/adj (the reciprocal of cosine)
cosecant: csc theta = hyp/opp (the reciprocal of sine)

The prefix "co-" is short for complementary. If we look at the other acute angle (90°-theta), the value for cosine of that angle will be the value for sine of theta, the complementary angle. This is because when we look at the triangle from the point of view of the other angle, the side we had labeled the opposite is now the adjacent and vice versa. The long side is the hypotenuse and it does not change depending on our point of view.

Besides the relationship of complementary values, there are several important trigonometric identities based on The Pythagorean Theorem, which we will discuss tomorrow.