Preliminaries

Circle

It all starts with circles! Ancient humans had straight lines, of course, but they also understood circles. Making a circle from a straight line is a brilliant invention, though quite easy in retrospect.

Question

  • Show how to make a circle from a straight line.
  • What is a unit circle?

Answer

Take a rope or a stick. Plant one end firmly, but so you can still rotate the rope or stick. Then keep it stretched, and take the other end for a walk, around. There, you traced a circle! (Oh, why did I not live 10,000 years ago? I could have been known for discovering this!)

drawing a circle

A unit circle is the circle obtained from a stick whose length - the circle’s diameter - is 1.

Angles

An angle (often denoted by \(\theta\)) is a measure of turn, and can be described in degrees or radians. Imagine you were looking straight ahead and then turn your head to look at something on the side. If you can keep on turning, and if you can turn all the way to where you were originally looking (straight ahead), that is considered a full turn, 360 degrees. Don’t try this, though, it could hurt your neck. Owls can rotate their head almost a full circle. Yash Shah can do almost 180 degrees.

Question

What kind of an angle do you get with 90 degrees (what is it called)? And how many degrees in a full turn?

Answer

360 degrees. So, one degree is 1/360th of a full turn.

It is useful to break the 360 degrees into 4 parts of 90 degrees – and a 90 degree turn makes a “right angle”. So, another way to think of one degree is 1/90th part of 90 degrees.

Triangles

A polygon is an enclosed geometric shape consisting of straight lines, vertices (points where the lines meet), angles, and no break in its perimeter.

A polygon with 3 sides is a triangle. And it has 3 angles.

The 3 angles of a triangle add up to 180 degrees.

Triangles

Radians vs Degrees

Now, to see what is a radian measure of an angle, pretend that you’re walking along the curve of the circle, starting from some point (say, at the base of the circle). If you walk exactly a distance equal to one radius length, then the angle covered while walking this distance is a radian (see the picture on the LEFT above). One radian.

Question

  • How many radians equals a degree? How many degrees is one radian?

  • Convert \(\frac{\pi}{4}\) radians to degrees.

  • Convert 90 degrees to radians.

Answer

  • Method: To convert between degrees and radians: Let’s call a unit radial angle as \(\theta\) in degrees.

    • How many such \(\theta\)’s in a full circle? A full circle is 360 degrees, so the answer is \(\frac{360}{\theta}\) of such angles to make a full circle.

    • Similarly, how many radians in a full-circle walk? We know the distance covered in the full circle is \(2 \pi r\) (where \(r\) is the radius, and also the unit length we’re looking at here).

    • Hence, we need \(2 \pi\) radians to make a full circle. Therefore, \(2 \pi = \frac{360}{\theta}\) and hence \(\theta = \frac{360}{2 \pi} = \frac{180}{\pi}\).

    • So, 1 radian = \(\frac{180}{\pi}\) degrees (about 57.29578). And 1 degree = \(\frac{\pi}{180}\) radians (about 0.0174533).

  • For instance, \(\frac{\pi}{4}\) radians equals 45 degrees.

  • And 90 degrees equals \(\frac{\pi}{2}\) radians.

Historical Facts and Primitives

We used two facts in the above analysis, is that a circle has 360 degrees, and two that its perimeter or circumference is \(2\pi r\). What are the raw or primitive sources of these facts?

Question

  • Why 360 degrees in a circle?
  • Why is its circumference equal to \(2 \pi r\)?

Answer

6,000 years ago, the Mesopotamians had created a base-60 number system. (See, besides 10, you also have another special number, 6 or 12. 60 minutes in an hour. 24 hours in a day.) Meanwhile, one of the early calendars at the time had 360 days. (Pretty accurate, no?)

Pizza

They passed on their love for 360 onto the early Egyptians, who decided that a circle should be split into 360 degrees. See, 360 = 60×6! 6 equal slices in a pizza, each a nearly–perfect equilateral triangle. Each with 60 degrees.

So, there, one mystery solved!

For, the second one, there is a straightforward answer. A long time ago, people wondered what was the distance traveled if you walked around a circle.

circumference

\(\pi\) is in fact * defined* as the ratio of the Circumference (C) of a circle to its diameter (D). So, \(\pi = \frac{C}{D} = \frac{C}{2 r}\). This was also decided long before my time.

However, they did not quite know what this value \(\frac{C}{D}\) was. This Greek fella thought it was a little more than 3. There is a very easy explanation for that. Figure it out? (Hint: the pizza with 6 slices?) The Greek fella became famous, you won’t because you were born too late. (OK, he also did many other cool things, plus he showed that \(\pi\) was not just greater than 3, but greater than \(3\frac{10}{71}\), though honestly that part required more muscle than brain.)

Functions

A function is a mapping between one set (S1, called DOMAIN) of items and a second set of items (S2, called RANGE), in such a way that an S1 item has a unique (exactly one) counterpart in S2. So, the second set has as many as, or fewer, items as the first set. A simple function is the mapping between state-names and their two-letter codes (or the other way round). Here’s another one.

Function maps Domain values to Range values

Two other functions we just defined were the above: converting from degrees to radians, and from radians to degrees. Let’s call the first function \(f\) and the second function \(g\) Then, the mathematical way to write these functions is

  • \(f(d) = \frac{\pi}{180} d\); (give me an angle in \(d\) degrees, and I’ll tell you it is \(f(d)\) radians)

  • \(g(r) = \frac{180}{\pi} r\); (give me an angle in \(r\) radians, and I’ll tell you it is \(g(r)\) degrees)

Functions can have one “argument” or multiple arguments. An argument is the \(x\) in \(f(x)\) – the thing or things that are needed to fully define the function. The above examples are all functions of one argument.

A simple example of a 2-argument function is a function that computes area of a rectange. The arguments are (w, h) – for width and height – and the function is \(f(w,h) = w *h\).

Question

Write the mathematical form of the function, and identify how many arguments it has: for a function that computes

  • the area of a square?
  • the area of an ellipse?

Answer

1 for square (\(f(x) = x^2\); where \(x\) is the length of either side), and 2 for ellipse (\(f(a,b) = \pi a b\); where \(a\) and \(b\) are the radii of the ellipse).

Graphs

A graph is a visual way of depicting a function. A basic graph in the x-y plane represents a single-argument function, where the x-value is the argument (input) and the y-value is the function’s output. Let’s consider the two functions that convert from radian to degrees and from degrees to radians.

The two functions look very similar, don’t they. But notice the x and y axes, and their ranges. What if we ensure that both axes have the same range (from 0 to 360). How would the functions look like then?

Trigonometric Functions

Suppose we want to trace a dot around a unit circle