As we talk and learn about trigonometry, keep one thing in mind: trigonometry is easy. In fact, I’d argue that the portion of trigonometry that we will cover in this article can be understood intuitively. And soon you shall.

We begin our task by considering the following standard Cartesian coordinate system.

Place a circle with radius 1 at the center of the graph, such that the circle’s center is aligned with the origin of the coordinate plane. This circle is called the unit circle.

Let’s see this circle in greater detail by zooming in on it.

The line that we draw is called the terminal side of our angle. All angles are measure from their initial side (the x-axis) to their terminal side. The angle created in the diagram below is 45 degrees. The green line below intersects the circumference of our circle at a specific point on the coordinate system.

This is true for any line drawn from the origin through any position on the circumference of the circle, as illustrated in the next diagram.

The upshot of all this is that, no matter the angle, given a point along the circumference of a unit circle, we can construct a right triangle, like so.

The terminal side of our angle forms the hypotenuse of our right triangle, and is always equal to 1, as mentioned earlier. The point at which our line crosses the circumference of the circle actually gives us the lengths of the remaining two legs. The x-value of our point equals the length of the leg that lies along the x-axis (“side x”). The y-value of our point corresponds to the length of the leg that lies parallel to the y-axis (‘”side y”).

Armed with the lengths of each leg of our triangle, it’s time to learn about two trigonometrical functions, sine and cosine. These functions will allow us to extract the information necessary to draw objects moving at any angle we want. Both functions take as their argument an angle, but in practice this can be any real number.

Side x of the triangle indicates how much horizontal distance is composed within the hypotenuse. This value is given to us by the cosine function, abbreviated in algebra as “cos”. The y side of the triangle indicates how much vertical distance is composed within the hypotenuse. This value is given to us by the sine function, abbreviated in algebra as “sin”. Together, these values tell us the amount of vertical movement needed and the amount of horizontal movement needed to move at the desired angle for 1 unit. “1 unit” in our case is “1 pixel”.

The sine and cosine functions return numbers between -1 and 1, inclusive. Because of this, using sine and cosine gives us the core of what matters in all this — the direction of the angle without any significant distance (we can’t well represent 77% of one pixel!). Once we have our direction, we can multiply the factors given to us by sine and cosine functions to draw lines.

There are two caveats. First, you must rewrite the expression in terms of simple functions. The difficulty of this task increases as the function gets larger, I would imagine. Second, I do not know if this technique works at levels calculus and above.

Click the picture below to enlarge it.

If you find yourself being asked to “find the average rate of change of function” and that function happens to be non-linear (e.g. x^2), you’re probably being asked to find the average rate of change of the secant line of that same function (x^2). I say this because it simply doesn’t make sense to find the average rate of change for a function that does not change at a constant rate.

Worth your consideration when dealing with average rates of change for any function in two variables: if you are given two x-values, you actually have two points. Place each x-value into the function and solve for y to get these points.

With two points in hand, your goal is to write the slope-intercept form of the secant line of the given function between the the two points you calculated above. (Recall how to do this? Given points P1 and P2, find the slope. Given P1 and the slope, write the equation of the line in point-slop format, then evaluate this equation to produce the slope-intercept form.)

The secant line is just like any other line, except the points that form a secant line lie on the graph of some function. In pre-calculus, the secant line isn’t of particular interest, as the average rate of change of a secant line can be manipulated to read zero for a graph that clearly is never constant (e.g. x^2 with P1 = (4,4) and P2 = (-4,4)). However, I believe that doing such a thing is actually an obvious misuse of the idea behind a secant line. To the best of my knowledge, the secant line is intended to tell you the instantaneous rate of change at one point along a curve.

It may confuse you to consider the purpose of the secant line when dealing with average rate of change of a non-linear function.

As an aside, consider this: the secant line of a linear function is the graph of the function itself.

Here’s a completely worked out example of finding the secant line of a non-linear function given two x-values and a function f.

Ponder this: what would be the average rate of change of f(x) = x^2 if you are given the following two x-values: -4, 4?

Family is one measure of success.

The number of beautiful, expensive prostitutes you have gleefully fucked is a measure of success.

Now it’s your turn. Define at least one measure of success, for you, personally, and add it to that list.

From that list, pick any number of measures you find desirable.

Congratulations. You have defined your personal definition of success.

Until you are able to measure success, you cannot know whether you are successful.

That’s my “motivating speaker” bit. Now for the math bit.

A ratio contains two parts. A numerator and a denominator. The denominator can be thought of as your base, or rate. An example of a rate in action can be seen by simply counting. You can count by 1: 1, 2, 3, 4… In each iteration, you counted at a rate of 1. Let’s consider a rate of 4: 4, 8, 12, 16… In each iteration, you counted at a rate of 4. The job of the denominator is to determine how many times (or what percentage) of its own area will it take to completely encompass the area of the numerator.

Another take: If I have one bottle that holds one gallon of water, will it be less than full, exactly full, or more than full (overflowing) if I pour second bottle of water into the first? Clearly, you cannot answer that question until you know the size of the second bottle of water. And that, is the numerator’s job.

A denominator could also be, say, American dollars. Simply saying, “I have $420.” could be written as 420/$. See it now? Denominators can be anything. Literally. Anything, except, that is, nothing.

Nothing is what you cannot measure.

That phrase again.

If we use the denominator as a way to measure something, ask yourself: how can I measure something (the numerator) based on nothing (the denominator)?

More than anything, I hope you take away from this a better understanding of what a ratio is and, possibly to a lesser extent, what division is. I also hope the answer will make the “undefined” definition of division by zero more clear to you.

Happy counting.