\( \definecolor{flatred}{RGB}{192, 57, 43} \definecolor{flatblue}{RGB}{41, 128, 185} \definecolor{flatgreen}{RGB}{39, 174, 96} \definecolor{flatpurple}{RGB}{142, 68, 173} \definecolor{flatlightblue}{RGB}{52, 152, 219} \definecolor{flatlightred}{RGB}{231, 76, 60} \definecolor{flatlightgreen}{RGB}{46, 204, 113} \definecolor{flatteal}{RGB}{22, 160, 133} \definecolor{flatpink}{RGB}{243, 104, 224}\)

February 12, 2021

The Circular Functions

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Textbook Reference: Section 5.2 Unit Circle: Sine and Cosine Functions

I first saw this brilliant way of introducing the circular functions because of James Tanton, the "Mathematician-at-Large" for the Mathematical Association of America, so a big thanks goes to him. You're welcome to watch his video on YouTube to see his own presentation of it.

Imagine you're in 5th century India. Back then you wouldn't have had a cell phone and Facebook, so you might have had a lot of time to just sit around and ponder the wonders of the universe we live in — wonders such as the sun. You might find yourself wondering:

Just what is that big beautiful ball in the sky we call the "sun"?

How far away is it?

How high up is it?

It's worth noting that a lot of the sort of math we're learning this semester was motivated by astronomy. People from many different cultures kept track of the motion of the sun, moon, stars, and planets, for both practical and philosophical reasons. We're concentrating on India in particular because the math they figured out is what leads to the idea of circular functions as we know them today.

Anyway, let's get back to figuring out where the sun is. Assuming the sun goes in a (roughly) circular path, it would always be the same radius away from the center of the earth. If you could figure out that radius, then you could do a little geometry to figure out how high the sun is based on your angle. To make things simple, you decide that the distance from the earth to the sun should always be \(1\). That way, if years down the line scientists were ever able to actually figure out that distance, all it takes is a quick multiplication and all your calculations will be accurate! (In fact, scientists today do call this distance \(1\) AU, which is short for astronomical unit.)

Likewise, to figure out how high up the sun is, you can't just go up there and drop a really long rope down as an altitude. The best you can do from your safe vantage point on the earth is measure the angle of elevation as you look up at it (not too directly, of course). So, at this point you want to find out a relationship between the angle and the height, watching how those quantities change as the sun rises in the east and sets in the west. While you're at it, you might as well also try to figure out how far left or right the sun is — its "overness", so to speak.

And that's exactly what they did in India — they developed a whole bunch of mathematics to figure out the height and overness of various angles, writing all sorts of manuscripts in Sanskrit and pushing the boundaries of mathematical knowledge at the time.


From here, the story gets pretty interesting as different cultures passed around this knowledge.

In the 9th and 10th centuries, Persian scholars came along these Sanskrit manuscripts with all sorts of interesting techniques for analyzing circles, and they decided to translate the manuscripts into Arabic. The Sanskrit word used for the height was "jya" (which comes from the word for a bow-and-arrow, which you might to see if you extend the height downward to the other side of the circle) or "jiva". The Persian scholars liked the second word, so they didn't try to translate it, instead deciding to just transcribe it as closely as possible from the Sanskrit (much like how when we talk about the famous Japanese food "寿司", we don't translate it as "vinegared rice" but just call it "sushi.") This gave them the word "jiba" — except that in Semitic languages like Arabic and Hebrew, writing vowels is optional, so instead of "jiba" they really just wrote "jb".

Now fast-forward to the 13th century, when European scholars came across these Arabic manuscripts and started to translate them into Latin. They got really confused when they saw "jb", because the only Arabic word they knew with those consonants was "jaib," which means a cavity or a bay — kind of a strange word to use for the height of the sun. Nevertheless, they went with it, and translated it to the Latin word for a cavity or bay: "sinus." Many countries today still call this height the sinus, but in English it got modified just a bit more to become what we know today as the sine of an angle.

In other words, our word sine has been translated from Sanskrit, to Arabic, to bad Arabic, to Latin, and finally to English. Yikes.

Height and Overness

"Wait, so what's the point of this history lesson?"

Because now we know what the sine is supposed to mean:

The sine of an angle
is its height on the unit circle.

And you can remember the mental image of finding the height of the sun.

Now, what about the overness?

Well, the overness of a particular angle can be drawn with a very similar picture to the height, but instead using the complementary angle:

In Latin this was written as "complementi sinus" (meaning "sine of the complement"), and was eventually abbreviated to "co.sinus."

As you might guess, nowadays we call it the "cosine".

So now we have another useful intuition:

The cosine of an angle
is its overness on the unit circle.

And you can remember the mental image of finding the overness of the sun.

What we've just introduced here are the primary object of study this semester, the circular functions, the sine (written \(\color{flatblue}\sin(\theta)\)) and the cosine (written \(\color{flatred}\cos(\theta)\)). Essentially, if you input any angle, they will tell you the height and overness respectively.


Let's look what happens when the sun first rises in the east (at \(0^\circ\) or \(0\) radians).

At this point, the sun is all the way over to the right, so the overness is \(1\). It hasn't risen at all in the sky, so its height is \(0\). In modern mathematical notation, we would say that \(\color{flatred}\cos(0^\circ)=1\) and \(\color{flatblue}\sin(0^\circ)=0\), and so the \(({\color{flatred}x},{\color{flatblue}y})\) coordinates of the point corresponding to \(0^\circ\) are \(({\color{flatred}1},{\color{flatblue}0})\).


Here's the big take-away:

For any angle \(\color{flatpurple}\theta\),
the corresponding point
on the unit circle is

Drag the sun around to change the diagram.
Click on the sun to turn its coordinates on or off.

In class, you'll be working to figure out the sine and cosine of various nice angles, based on the properties of particularly "nice" right triangles.

Preview Activity 2

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

  1. What are the height and overness of the sun when it's directly overhead at noon?
    How would you write this in modern mathematical notation?
  2. What are the height and overness of the sun when it sets in the west?
    How would you write this in modern mathematical notation?
  3. What are the height and overness of the sun at midnight (assuming it's directly under where you are on the earth)?
    How would you write this in modern mathematical notation?
  4. What are the height and overness of the sun when it makes a \(\color{flatpurple}32^\circ\) angle of elevation?
    How would you write this in modern mathematical notation?
    (You'll need to use a calculator, or you can use the interactive diagram above.)
  5. At how many times during the day is the height of the sun equal to \(\color{flatblue}\dfrac12\)? What are the angles at which this occurs?
  6. Answer AT LEAST one of the following questions:
    1. What was something you found interesting about this reading?
    2. What was an "a-ha" moment you had while doing this reading?
    3. What was the muddiest point of this reading for you?
    4. What question(s) do you have about anything you've read?


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