Lessons 14-3 and 14-4: The Sine, Cosine, and Tangent Ratios (Day 144)

This is what Theoni Pappas writes on page 121 of her Magic of Mathematics:

"Bees...by virtue of a certain geometrical forethought...know that the hexagon is greater than the square and the triangle, and will hold more honey for the same expenditure of material. -- Pappus of Alexandria."

So here Pappas is quoting Pappus -- that is, Theoni Pappas is quoting the fourth century Greek mathematician Pappus of Alexandria. And I did say that I'd be writing what Pappas says about the birds and the bees -- honeybees, that is.

In this section, "What Honeybees Are Buzzing About Mathematics," Pappas (and Pappus) will answer the age-old question, why are honeycombs hexagonal? We will discuss their answer over the next few days.

Meanwhile, this week I checked out a movie from the library, a movie that was released last year, but I never watched it until now. It's called The Man Who Knew Infinity, and its title character is the early twentieth century Indian mathematician Srinivasa Ramanujan. The movie details the friendship he had with the British mathematician G.H. Hardy.

That's right -- there were two math films released last year. But Hidden Figures received much more attention than The Man Who Knew Infinity due to the former's Oscar nominations. I must point out that both Ramanujan and Katherine Johnson faced much prejudice as they performed calculations.

About ten years ago, there was a TV show called Numb3rs. The main character was a mathematician who helped his brother, an FBI agent, solve crimes. As it turns out, the main love interest on the show was a South Asian grad student named Amita Ramanujan -- and of course, the writers named the character after the famous mathematician.

As usual, I don't have time now to write more about the movie on the blog, but I will watch it at some point this week.

This is what I wrote last year about today's topic:

Lesson 14-3 of the U of Chicago text is on the tangent ratio, and Lesson 14-4 of the U of Chicago text is on the sine and cosine ratios. I have decided to combine all three trig ratios into one lesson.

David Joyce was not too thrilled to have trig in the geometry course. He wrote:

Chapter 11 [of the Prentice-Hall text -- dw] covers right-triangle trigonometry. It's hard to see how there's any time left for trigonometry in a course on geometry, but at least it should be possible to prove the basic facts of trigonometry once the theory of similar triangles is done. The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?) The one theorem of the chapter (area of triangle = 1/2 bc sin A) is given for acute triangles.

As the trig functions for obtuse angles aren't covered, and applications of trig to non-right triangles aren't mentioned, it would probably be better to remove this chapter entirely.

Yet most geometry books include trig because most state standards require it. And this most certainly includes the Common Core Standards:

Define trigonometric ratios and solve problems involving right triangles

CCSS.MATH.CONTENT.HSG.SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

CCSS.MATH.CONTENT.HSG.SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles.

CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

And all three of these standards appear in this lesson.

Some people may wonder, why do we use the same name "tangent" to refer to the "tangent" of a circle and the "tangent" in trigonometry? A Michigan math teacher, Mike Shelly, discusses the reasons at the following link:

http://blog.michaelshelly.org/2009/12/why-is-trigonometric-function-called.html

On the other hand, the reason that "sine" and "cosine" have the same name is less of a mystery. In fact, the U of Chicago tells us that "cosine" actually means complement's sine -- since the cosine of an angle is the sine of its complement. This is Common Core Standard C.7 above.

Last Friday's post was a whirlwind of ideas, and today's post continues these ideas. In the last two days, I linked to a variety of sources in search of answers to questions such as:

-- Should activities be taught during the trig unit?

-- Should a trig unit be taught during the Geometry class?

-- Should a Geometry class be taught during high school?

We searched high and low, from traditionalists to their opponents, seeking these answers. I fear that when I post links to all these competing sources, my own opinions are obscured. The blog readers know what David Joyce and the traditionalists believe, but not what I myself believe.

Well, here's my belief -- I answer all three of those questions in the affirmative. High school should have a Geometry class, Geometry class should have a trig unit, and a trig unit should have activities -- and I posted my activity for the trig unit of a high school Geometry course yesterday.

I also think back to the activity that sparked this debate -- proofs and the courtroom. We saw how the traditionalists objected to the courtroom activity on the grounds that it is too long.

I admit that I'm fascinated with the idea of using a courtroom to highlight Geometry proofs. I took Geometry back during the 1994-95 school year -- the year of the famous OJ Simpson trial. And so I often fantasized that my Geometry class was a courtroom -- the People's Court. Actually, that TV show was off the air during that year. But it made a comeback in 1996, the first full year after the Simpson trial, as TV stations were trying to capitalize on the Simpson trial's popularity. (This was the same year that another famous courtroom show debuted -- Judge Judy.)

So I might organize a People's Court during my Geometry classes. When I would teach the lesson depends on what textbook I was using. If I had Michael Serra's text, People's Court would occur at the end of the year, around Chapter 13. With the U of Chicago text, court may occur in Chapter 3 (when the class first learns about proofs), and in many other texts, it may occur in Chapter 4 (where triangle congruence proofs appear).

One way to prevent the unit from taking too long is to assign each group a different medium-level proof -- then they present those proofs when the class actually reaches that unit! So one group may be assigned the Isosceles Triangle Theorem to put on trial a week later, while another is assigned some of the Parallelogram Theorems to put on trial a few months later. As long as all groups present before the end of the first semester, it works out in the end.

By the way, many of our students may have trouble with trigonometry, but one youngster who had no trouble with trig was -- you guessed it -- Ramanujan. When he was in the equivalent of seventh grade, an older friend lent him a trig text book, and the young genius mastered it that year!

Here's one more connection between Ramanujan and trig. The U of Chicago text tells us how to find some trig values exactly, but not others. For example, cos(60) = 1/2, but cos(20), cos(40), and cos(80) aren't as easy to find. Well, the Indian genius found an interesting formula connecting the three cosines whose values we can't find. (All values are in degrees -- "cbrt" is cube root.)

cbrt(cos(40)) + cbrt(cos(80)) - cbrt(cos(20)) = cbrt(1.5(cbrt(9) - 2))

A 20-degree angle is not constructible and so its cosine can't be written exactly using square or cube roots -- of real numbers, that is. Complex numbers are a different issue:

cos(20) = (cbrt(a) + cbrt(b))/2

where a and b are the complex cube roots of 1 -- that is, a = (1 + i sqrt(3))/2, b = (1 - i sqrt(3))/2.

Now you can see why we have high school students memorize cos(60) and not cos(20).

OK, let me post the worksheet: