Introduction

This tutorial is meant as a sequel to my unit circle tutorial. Much of this tutorial assumes you have already been through that one, so if haven't already done so, please go through the previous tutorial now.

The unit circle is meant as a sort of idealized circle, from which all other measurements can be scaled up. Radians, for example, made it easy to determine how far a wheel was traveling, given its rotation in radians. All that was required was a single multiplication.

As a matter of fact, all the measurements on the unit circle work this way. Sine and cosine can be applied to real world measurements with a single multiplication to scale them up or down.

That's why, in the unit circle, all the trigonometric functions do a sort of double duty. They're ratios, as you've been taught, but in the idealized world of the unit circle, they can also be treated as absolutes, since they result in useful coordinates. For example, sine and cosine are both ratios that relate in different ways to the length of the hypotenuse, but also give exact coordinates to graph.

In this tutorial, we'll go beyond sine and cosine, and explore the other trigonometric functions of tangent, cotangent, secant, and cosecant. The goal of this tutorial is to explain them in a way that can be easily understood, comprehended, and remembered.

To explain these, we're going to scale the triangle up beyond the bounds of the unit circle. In the previous tutorial, the hypotenuse always had a length of 1 because it was representing the radius of the unit circle.

The hypotenuse of a right triangle is, by definition, the longest of the three sides, so this meant that, with any given angle, the other two side must always fall on or inside the unit circle itself, because they have to be 1 (in the case of 90 degree or π/2 angles) or less.

What happens, though, if we scale up a right triangle in the unit circle so that the width or height is 1 instead of the hypotenuse? In this tutorial, we'll try that out, and examine the useful measurements that result.

To start off simply, however, we'll focus on the one thing that doesn't change as the right triangle is scaled up - its slope.

Slope

Question: What's the slope of a 30 degree angle? Many would answer something like, “What kind of question is that? It's 30 degrees!“

I'd give that answer half credit. It does recognize that, regardless of the lengths of the sides of the triangle, the slope does remain the same.

However, slope is usually stated as decimal number, so that it's easy to multiply by. Here's a quick refresher course in slope, excerpted from the series The Mechanical Universe:


“Change in elevation” over “change in horizontal distance” is usually stated in a way that's catchier and easier to remember:



For example, take a 45 degree angle in the unit circle. When the hypotenuse was 1 (the radius of the unit circle), what was the rise and what was the run? If you remember the hand trick from the video, it shouldn't be too hard to recall:



So, for a 45-degree angle the slope is 1. In other words, for every 1 unit you move horizontally, you're going to move 1 unit vertically, as well. This makes sense for a 45-degree right triangle, since the two shorter sides are the same length.

What about that 30-degree angle I asked about earlier? Lets work through that problem:



You can see a more exact answer at Wolfram|Alpha. It can also be stated as the square root of 3 over 3.

As you'll see it again and again, you should find a way to remember the square root of 3.

The number is less exact than in our 45-degree answer, but the meaning remains the same. With a 30-degree angle, for every 1 unit you run (move horizontally), you're going to rise (move vertically) 0.57735... units. That's more than half a unit, but less than a slope of 0.60.

We've been talking quite a bit about slope, but not much about the unit circle here. This is a very important tangent however, as you'll learn in the next section.

Tangent

Ever been told by someone that you're going off on a tangent? Ultimately, the complaint is that you're going off on a line that will only take the discussion farther and farther from the main point.

It's almost exactly the same definition in math. One mathematical definition of tangent is a line that touches a circle (a unit circle, for our purposes) at only one point. Starting at that one point, and then traveling along the tangent would only take you farther and farther away from that main point.



As you can see, a tangent is easy to draw. Draw a radius line at any angle, and at the point where it touches the circle, draw a line perpendicular (at a 90° angle) to the radius, and that's the tangent.

To keep things simple in this tutorial, we're only going to consider the horizontal and vertical tangents. In this particular section, we'll only focus on the vertical tangent.

The formula to graph a vertical tangent is easy, it's x=1. When y=0, x=1. When y=5, x=1. When y=4,287, x=1. You get the idea. Not surprisingly, the only point at which it intersects the unit circle is

In the previous tutorial, we always created the length of the hypotenuse (the longest side) to be 1. Imagine that, instead of the hypotenuse having length 1, we had the base have a length of 1.

Here's a picture of the situation:



Here's a question for you: If we tried scaling up a 45° right triangle in this manner, what would the coordinates be where the hypotenuse of this new larger triangle intersects the tangent line?

Let's think about this. Because the tangent is defined as x=1, the x coordinate where the hypotenuse intersects the tangent will also obviously be x=1. So, the coordinates we have so far are (1, something). That's half the work already done!

In the previous section, we worked out that the slope of a 45° angle was 1, at that meant that for every 1 unit you moved horizontally, you moved 1 unit vertically. As it happens, a base width of 1 means we are moving over 1 unit horizontally, so we should obviously move up 1 unit vertically!

So, the coordinates of a 45° angle where it intersects the tangent line would simply by (1,1)!

Did that seem easier than it should have? Let's try it with our 30° angle example, too. What was the slope of that angle? It was 0.57735... and so on. Let's take a look at that angle plotted by Wolfram|Alpha.

That means, for every 1 unit we travel horizontally, we travel 0.57735... units vertically. Yep, the coordinates where the hypotenuse meets the tangent line is (1, 0.57735...).

See the pattern? Whenever a right triangle with a given angle has a base length of 1 in the unit circle, the coordinates where the hypotenuse will intersect the vertical tangent line is (1, slope of that angle)!

As with sine and cosine in the unit circle, the slope is doing double duty. It's both the slope itself, and the y coordinate where the hypotenuse intersects the tangent at x=1. In fact, we could just call this number the tangent.

Remember SOHCAHTOA? That helps remind us, among other things that the formula for the tangent is the opposite side's length over the adjacent side's length. Also, remember that those lengths on the unit circle are worked out by figuring sine (the height) and the cosine (the width)

Let's take a closer look at the formulas for slope and tangent:



Yep, the tangent and the slope are always the same, which is why it works out as it does! This should also help you better understand the brief reference to figuring out tangent on your fingers from the previous tutorial.

So far, we've only dealt with scaling triangles up to meet a vertical tangent. What happens if we scale them up to meet a horizontal tangent?

Horizontal Tangent

With a horizontal tangent, we're simply dealing with what happens when y=1, instead of x=1.

Remember how we determined the tangent (the same as the slope) of a 45° angle was 1, so the coordinates where the hypotenuse met the vertical tangent was (1,1)? For a 45°, this works out nicely, since it intersects the horizontal tangent in the same place.

Let's go back to our 30° example, and get a better idea of what changes. The slope is 0.57735..., so let's view what happens when we try this out:



Here's another look at the same situation, via Wolfram|Alpha (albeit slightly distorted). Even without knowing the exact coordinates, we can see that the hypotenuse, in the 30° case, is MUCH longer than where it intersected the vertical tangent.

So, what are the coordinates? We start in a manner similar to before, with the coordinates (something, 1), because the horizontal tangent formula is y=1 (instead of x=1 for the vertical tangent).

In the case of the vertical tangent, we just multiplied 1 (the x coordinate) times the slope to get the y coordinate. Since we're still talking about the same 30° angle, the slope is still the same 0.57735... we used before. We'll just have to come at it from the other way:



You can actually see several important things here. First, we found the coordinates for where our 30° angle intersects the horizontal tangent at (1.73205...,1). Second, while the tangent relationship of y = x × 0.57735... still holds, it would seem easier just to state the relationship the other way around, as in x = y × 1.73205..., especially when that lets you multiply by 1.

Finally, note that you can always find where the hypotenuse intersects the horizontal tangent by dividing 1 by the tangent of the same angle. That's why this number is given the name cotangent.

If you think about it, there are several ways to find the cotangent:



As a brief review, we covered sine and cosine in the previous tutorial, and now we've covered tangent and cotangent. If you think of the horizontal tangent as being a cotangent line, then this is easier to understand.

We keep talking about the slope of the hypotenuse and the coordinates where the hypotenuse intersects the tangent and cotangent lines, but what happens to the length of the hypotenuse as triangles are scaled up to meet the tangent and cotangent lines? That's discussed in the next section.

Hypotenuse Length

We're going to switch from focusing on the slope of the hypotenuse to its length.

This almost means a return to our old friend from the right-triangle, the Pythagorean Theorem: a² + b² = c².

When the hypotenuse is 1 unit long, the squares of the other two sides must add up to 1. For example, in our 30° angle, where the cosine (width) is 0.8660... and the sine (height) is 0.5, we get:



However, the hypotenuse must get longer to meet the tangent lines as we've seen. Let's start as before, scaling up to meet the tangent line (that's the vertical tangent). We've already seen that the coordinates of a 30° angle gives us coordinates of (1, 0.57735...). As in the above example, we note that the coordinates are also the lengths of the two smaller sides. That being the case, let's figure out how long the hypotenuse is when it meets the vertical tangent:



So, we see that the hypotenuse is now 1.1547... units long. Since the original hypotenuse was only 1 unit long, this is a sort of scaling factor for the hypotenuse.

Take another look, though. When we scaled the width of a 30° right triangle from 0.866... to 1, that's a factor of 1.1547... times, as well. The height went from 0.5 units to 0.57735... units, which is also a scaling factor of 1.1547... times!

So, this one factor, when scaling a right triangle up to meet the tangent line tells us how to scale the lengths of all the sides in order to do so! Since we're scaling up from a hypotenuse of 1, this factor also gives us the exact length of the hypotenuse after being scaled up to the tangent line.

In trigonometry, this factor is called the secant. Since we scale the width of the triangle (the cosine, in the unit triangle) up to 1, it shouldn't be surprising that we can find the secant in this manner:



The hypotenuse length was 1 before scaling it up, so you could also work it out by dividing the hypotenuse by the width:

Meeting the Cotangent Line

Let's not forget the length of hypotenuse when scaled up to meet the cotangent line (that's the horizontal tangent line).

Our example 30° right triangle, when scaled up to meet the cotangent line, we found the coordinates (and side lengths) of (1.73205...,1). Let's run the numbers just as before, and find the hypotenuse length:



Oh! The hypotenuse in this case is exactly 2 units long. We took the height (the sine) from 0.5 to 1, which is a factor of exactly 2 units, so we shouldn't be surprised when the hypotenuse (and the width, for that matter) scales up by a factor of 2 units.

Since this scaling factor deals with scaling the height up to the cotangent line, this factor is naturally called the cosecant. The formulas for cosecant are as follows:



At this point, you should understand all 6 trigonometric functions: sine, cosine, tangent, cotangent, secant and cosecant. Once understood, however, they can be easily confused. In our final section, I'll offer some mnemonics to help keep them all straight.

Mnemonics

When you have the explanations and formulas in front of you, it's much easier to keep these things straight. How is it possible to keep all this straight in your head?

Let's break things up several ways. First, by the pairs in which you learned them.

Sine and cosine can be thought of as the simple lengths of the sides in the unit circle, when the hypotenuse is 1. Think of signs that tell you the length of a road.

Tangent and cotangent deal with the unit coordinates when the triangles are scaled up to meet the tangent and the cotangent lines respectively. This is its own mnemonic: tangents deal with tangents.

Secant and cosecant are the scaling factors to the tangent and cotangent lines respectively. Think secant means “secaling” factor.

Once you get the functions themselves straight, it can seem tricky to remember which of each pair deals with horizontal information and which ones deal with vertical information. However, there's an almost built-in mnemonic: look for the “o”.

Sine is the vertical length of the triangle. Tangent deals with coordinates when scaled up to the vertical tangent line. Secant deals with scaling factors up to the vertical tangent.

On the other hand, cosine is the horizontal length of the triangle. Cotangent deals with coordinates when scaled up to the horizontal tangent line. Cosecant deals with scaling factors up to the horizontal tangent.

Do you see what I mean by “looking for the o”? The words horizontal, cosine, cotangent, and cosecant all have an o in them, so they all go together.

Similarly, none of the words vertical, sine, tangent, and secant feature an o in them, so they all go together.

Looking for the letter o is an appropriate mnemonic for a circle, don't you think?

Another challenge is remembering the respective formulas. As mentioned before, SOHCAHTOA is the classic way to remember this, but that only gives 3 formulas.

Here's a lesser known yet more amusing mnemonic that gives all 6 formulas. Start by writing down the dividing lines, and writing “OOH AAH” above them:



“OOH AAH” is, no doubt, the sound you made upon first learning about unit circles. On the bottom, you're going to write “OOH AAH” again, but this time from right to left (as in “HAA HOO”):



From left to right, these are the formulas for the three vertical functions, sine (O/H), tangent (O/A), secant (H/A), followed by the three horizontal functions in the same order, cosine (A/H), cotangent (A/O), and cosecant (H/O).

You've also noticed that you can get some of these ratios simply by taking the inverse (dividing 1 by another number) of other factors. Tangent and cotangent are easy:



The ones beginning with t's are easy, so they can be thought of together. The ones featuring c's and s's are a little trickier:



When focusing on secant or cosecant, ignore the first s or c you see, and look for the second s or c. In the case of secant, you'd look at it as seCant. This C lets you know that it's the inverse of the function that begins with C - cosine!

With cosecant, you'd look at it as coSecant, and remember that this S it telling you that it's the inverse of the other S - sine!

Once you have that relationship down, here's a video that will take your knowledge of the trigonometric functions to the next level. It plays quickly, so you should pause it to read and understand everything it says:



Any further questions, class?

We'll start with the basics by introducting the concept of radians. What exactly are radians?

Radians are an alternative to degrees. Let's make sure that everyone is on the same page, with a quick and fun refresher course in degrees:



Most people are familiar with degrees. They're an absolute necessity when studying things like geography (where on Earth am I?) or astronomy (When will I be able to see a given star from where I am on Earth?). So why do we even need an alternative to degrees?

I'll highlight the problem with a smaller-scale example. Imagine a public park with circular running tracks of varying sizes, and two people, one who is trying to build his speed and stamina for a 100 meter race, and the other person who is timing the runner. They find a track, and the timer stands in the center of the circular track. The runner notes that the circular track is some weird amount, say, 142 meters in circumference, so he needs to know the location of the 100 meter mark.

The timer whips out his calculator, divides 100 by 142, getting 0.704225352. He multiplies this by 360 degrees, since the track is circular, and cheerfully exclaims that all the runner has to do is run 254 degrees (253.521127 degrees, to be more precise) to cover 100 meters on that track. At this point, the runner gives the timer a funny look.

See the problem now? 254 degrees is great when describing how far the timer, standing in the center of the circular track, has to turn to watch the runner. This doesn't help the runner much, as the runner isn't standing in the center. That's the whole thing with degrees; they're great for observers, such as the timer, but not so much for people or things moving around a circumference, such as our runner. This is why there's a need for an alternative to degrees.

Note that the absolute distance of 100 meters isn't of much help, either. On the different sized tracks in the park, that distance will be a different number of laps on each track. So, absolute distance and degrees aren't handy, thus we introduce radians. So what exactly are they?

Since the radius of a circle has a constant relationship with the circumference, that of Pi times twice the radius, how about counting off how far we've traveled around the circumference of a circle in terms of that circle's own “radius units”? It's a good idea, but “radius units” sounds weird (I think it's the repetition of the letter U), so we use the term radians instead!

It boils down to this: 1 radian is simply the length of a circle's radius traveled around that same circle's circumference.


That also gives us the formula to calculate radians: Radians = distance traveled / radius.

Math books like to scare you by writing this same formula as θ = s / r. θ is pronounced “theta”, and simply refers to the answer in radians. s means “arc length”, or, the distance traveled around the circumference. r, the only straightforward shorthand in this formula, means the length of the radius.

Going back briefly to the oberver's point of view, 1 radian translates into about 57.3 degrees (actually 57.29577951... and on and on). For the oberver, this certainly isn't a nice neat number, like 90 degrees, 180 degrees, or 360 degrees. As we've already learned though, the observer's viewpoint is not the point of radians.

However, in the next tab, we'll introduce the unit circle concept itself, and see how radians make things nice and easy for the person or thing that is moving around it.

Since we now have the scalable concept of radians to work with, we can now develop an entire scalable circle with which to work. We'll start by drawing on graph paper a circle centered at the (0,0) mark that has a 1 unit radius (remember cartesian coordinates?). It looks at it does below, with the coordinates marked where it crosses an axis:


That fact that it's a circle with a one-unit radius gives us the shorthand term for it: unit circle. Let's take a closer look at it what a unit circle can do.

Here's your first question: Since radians measure how far around a circle you've traveled, what's the distance around a full unit circle in radians? Let's work this out.

Our circle has a radius of 1 unit, so the circumference (our total distance traveled, in this case) is 2 times the radius times Pi, so we have 2 times 1 unit, giving 2 units, times Pi, resulting in 2 * Pi, or 2π for short (That π symbol is the Pi symbol, not a small letter n). That's just the total distance traveled, though.

To convert that to radians, we need to divide by the radius, of course. Our radius is 1 unit, so we work out 2π/1, which is simply 2π. In other words, the distance around a full circle is 2π radians in length!

Now you see why 1 radian is such a weird angle in degrees (1 radian = 57.29577951... degrees). When doubled and multiplied by Pi, it has to give a nice even 360 degrees. Again, we're going to stick to radians in this discussion, so just think of 2π radians as a full circle.

From here, it's not hard to see that π radians gives us half of a circle, and π/2 radians gives us a quarter circle. Three-quarters of a circle, then, would give us a distance of 3π/2 radians.


So, what distances are we traveling in radians when we traveling 45 degrees? 45 degrees is the centered observer's way of saying an eighth of a circle. The mover thinks of a circle as being 2π radians, so 2π times 1/8 would be 2π/8, or the same as π/4 radians.

Every 1/8 of a trip around the circle, then, would be in units of π/4 radians:

• 1/8 = π/4 radians
• 2/8 = 2π/4 = π/2 radians
• 3/8 = 3π/4 radians
• 4/8 = 4π/4 = &pi radians
• 5/8 = 5π/4 radians
• 6/8 = 6π/4 = 3π/2 radians
• 7/8 = 7π/4 radians
• 8/8 = 8π/4 = 2π radians

When you see the patterns and understand the process, it's actually not hard to understand.

Since angles of 30 degrees, or 1/12 of the distance around the circle, are also common, they're not hard to work out in radians. 1/12 of 2π radians comes down to π/6 radians. From there, we get:

• 1/12 = π/6 radians
• 2/12 = 2π/6 = π/3 radians
• 3/12 = 3π/6 = π/2 radians
• 4/12 = 4π/6 = 2π/3 radians
• 5/12 = 5π/6 radians
• 6/12 = 6π/6 = π radians
• 7/12 = 7π/6 radians
• 8/12 = 8π/6 = 4π/3 radians
• 9/12 = 9π/6 = 3π/2 radians
• 10/12 = 10π/6 = 5π/3 radians
• 11/12 = 11π/6 radians
• 12/12 = 12π/6 = 2π radians

Here's everything you've learned so far in one diagram, with the 90 degree units marked in black, the 45 degree units marked in red, and the 30 degree units marked in blue:


So, if you can remember that 2π radians takes you all the way around the circle, and that π radians, as well as 6π/6 and 4π/4 radians, takes you halfway around the circle, working out the rest of the numbers isn't difficult at all.

In the next tab, we'll discuss another amazing way the unit circle becomes useful when we focus on figuring out coordinates.

I have some quick refresher questions before we move on. When you've traveled π/2 radians, at what (x,y) coordinates are you? You're at (0,1).

Here's a tougher one: When you've traveled π/4 radians, at what (x,y) coordinates are you? That one's a little trickier. For this one, we're going to have to go back the viewpoint of the observer in the center, and thus back to degrees.

The radius of the unit circle is, of course, always 1 unit. If we construct a right triangle with the radius line as the hypotenuse, we could construct a right triangle for any angle in the circle:


Since π/4 radians from the central observer's point of view is 45 degrees, we'll create a 45 degree right triangle to help work out those coordinates:


Since we're looking for the (x,y) coordinate, the length of the side adjacent to our angle (the one running along the x axis) would give us the x coordinate. The height of the side opposite the angle (the one running straight up to meet the radius/hypotenuse) would give us the y coordinate.

If you remember all those lessons about right triangles, it's at this point where you begin to realize how many tools we can use here.

• Pythagorean Theorem: a² + b² = c²
• Sine of any angle = opposite/hypotenuse
• Cosine of any angle = adjacent/hypotenuse

(Don't forget: SOHCAHTOA)

Since the hypotenuse is 1, and both the sine and cosine divide by the hypotenuse, not only does this make the math nice and easy, but the answers will also be the exact coordinates we need!

The x coordinate is the length of the side adjacent to the angle, so we need to use the cosine formula. By cheating and using a calculator (make sure to set it in degrees, not radians!) we find that cos(45 degrees) = 0.707106781.... To get the y coordinate, we find the length of the side opposite the angle by using sin(45 degrees), which is also 0.707106781....

So, we have our coordinates for a 45 degree, or π/4 radian, angle: (0.707106781...,0.707106781...). Hmmm, that's another one of those numbers that goes on forever. Isn't there a better way to state that number?

Let's run through the Pythagorean Theorem approach, especially know that we know that sides a and b are equal, and see what we can come up with:



“OK,” you say, “so what? We just came up with the same numbers.” Let's back up a bit to to where a squared (and b squared, in the specific case of a 45 degree right triangle) was equal to 1/2. Instead of using .5, let's see where working with that 1/2 as a fraction can take us:



Hmmm...writing one over the square root of 2 is certainly much easier than that long string of numbers. Indeed, when you're tested on unit circles on many standardized tests, they'll usually ask you to write the π/4 radian or 45-degree coordinate in exactly that way.

Before moving on, though, I'd like to make that fraction a bit neater by making the bottom a whole number. We do that by multiplying both the top and bottom by the square root of two, and expressing the coordinates as fractions:



Not only is this a cleaner way to write the coordinates, it will make this and the other coordinates we work out much easier to remember, as you'll see in later sections.

The fractions may look funny, but all they're really saying is, “If you take the square root of 2, and divide it in half, that's a shortcut to working out this coordinate.” Or, put the opposite way, “Hey, if you take this coordinate, double it, and then square it, you get a nice, simple number, in this case - 2!”

We'll work out the coordinates and their corresponding fractions for the 30- and 60-degree angles in the next section, as well as their multiples.

So, now we not only know the coordinates for a 45 degree, or π/4 radian, point on a circle, we know that the cosine will give us the x coordinate and that the sine will give us the y coordinate on the unit circle. We also know that expressing these coordinates in fractions is much simpler than writing out the endless irrational decimals.

Let's work out the coordinates for the point at 30 degrees (π/6 radians):



Hey! That .5 is nice. That's easy to express as a fraction. But what about that other number? Well, everything else has been expressed over 2, so let's see what happens when we express that number over 2:



Ummm...yuck. Well, square roots seem to be popular in these fractions. Is 1.73205081... the square root of anything? Yep! It turns out that it's the square root of 3! So, the coordinates could be expressed this way:



Next, let's work out the coordinates for 60 degrees (π/3 radians):



Whoah! We get the same numbers as the 30 degree angle, only switched! That's not a coincidence. Remember that the internal angles of triangles always add up to 180 degrees. With a right triangle (a triangle with a 90 degree angle in it) that has a 30 degree angle in it, we find that 180 - 90 - 30 = 60 degrees, the remaining angle. Effectively, the 60 degree coordinates are just the 30 degree coordinates viewed from the other side of the triangle.

Let's review all the information, including the coordinates, that we know so far:


If you think of 1 as also being the square root of 1, you could write 1/2 this way:



So, at all the points where the circle crosses an axis, the coordinates involve 0, +1, or -1 - Simple! And at the oft-used 30° (π/6 radians), 45° (π/4 radians), and 60° (π/6 radians), all the coordinates involve these interesting fractions:



See that? Everything in the fractions is a square root of something over 2! Even better, there's a simple 1, 2, 3 progression! Even with all the complex math involved, it's all boiling down to 0, 1, 2, and 3.

That's easy enough to remember, but how do you remember which fractions are set at which coordinates, especially since it's not hard to make a mistake and switch around the coordinates for both 30° and 60°? Here's a video that will teach you quickly, using that 1, 2, 3 progression:



What about all the remaining major angles, such as 120°, 135°, and so on?

Well, you could go through and work out the coordinates laboriously as we did above, working out the sines and cosines, and so on. However, there's an easier way. We're measuring the 135° (3π/4 radians) angle from the 0° angle. When measured from the 180° (π radians) angle, it works out to be our old friend, the 45° (π/4 radians) angle.

One difference, though, is that we're now on the negative side of the x axis, so that the x coordinate will be negative. The means the coordinate for the 135° (3π/4 radians) angle will be:



Similarly, you can work out the 150° angle as a 30° angle, and so on. They work out to the coordinates we've already determined, but you need to make sure that you adjust the signs (+ or -) for the appropriate section.

Are you ready for this? Here's the entire unit circle worked out:


And here's how to draw it that so you remember all that data:



I like the approach used in this video of counting the π/3, π/4, and π/6 sections separately, so as to keep them mentally separate.

In the next tab, I'll review and break down the patterns for easier understanding. You'll also learn how to do some quick math on your fingers to get the coordinates.

It may seem like a lot that you've learned so far, but it boils down to a few basic things:

• Degrees deal with the observer's point of view. Radians deal with the mover's point of view.
• 1 radian = 1 radius length around the circle's circumference.
• Radians work very well with Pi, to the extent that π radians = 180 degrees, and 2π radians = 360 degrees.
• If you think of π radians as being 180 degrees, and you know your multiples of 30 and 45, the remaining degree angles aren't difficult to work out in radians.
• All the coordinate points where the circle cross an axis involve a 0, a -1, or a +1. Knowing coordinates makes it easy to work out which is which.
• All the coordinate points of π/6 (30°), π/4 (45°), and π/3 (60°) radians involve fractions easily remembered with the 1, 2, 3 progression taught in the video in the previous tab.
• The other major coordinate points can be worked out quite easily from those first 3 sets, as long as you adjust the signs properly.
• Cosine will always give the x coordinate on the unit circle, and sine will always give the y coordinate on the unit circle.

That last fact is especially interesting. Thanks to the simple progressions involved, you can actually work out the cosine and sine on your fingers with a simple trick! If you're familiar with tangents and cotangents, this video teaches how to handle those on your fingers, as well.



Betterexplained.com has a wonderful article you should read at this point, called Intuitive Guide to Angles, Degrees and Radians.

It really drives home the power of radians. Take this example from the article:

Let’s try a real example: you have a bus with wheels of radius 2 meters (it’s a monster truck bus). I’ll say how fast the wheels are turning and you say how fast the bus is moving. Ready?

“The wheels are turning 2000 degrees per second”. You’d think:

Ok, the wheels are going 2000 degrees per second. That means it’s turning 2000/360 or 5 and 5/9ths rotations per second. Circumference = 2 * pi * r, so it’s moving, um, 2 * 3.14 * 5 and 5/9ths… where’s my calculator…

“The wheels are turning 6 radians per second”. You’d think:

Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 * 2 = 12 meters per second. Next question.

Wow! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed. All because radians speak in terms of the mover.

The reverse is easy too. Suppose you’re cruising 90 feet per second on the highway (60 miles per hour) on your 24″ inch rims (radius 1 foot). How fast are the wheels turning?

Well, 90 feet per second / 1 foot radius = 90 radians per second.

That was easy. I suspect rappers sing about 24″ rims for this very reason.

I'm not sure that's the reason they sing about 24″ rims, but I'll go along with it while it lasts.

Think of the power you have here. Sure, the unit circle seems complex at first, but the power and patterns it presents when remembered and understood make many things simple.

Scaling up to real-world circles and back down to the unit of circles, as well as determining their motion, becomes a simple matter of multiplication. You can work out sines, cosines, tangents, and cotangents on your fingers.

Converting from radians to degrees and back, with a little practice, isn't difficult.

Which brings to mind the question of practice. The simplest practice I can offer is to print out several copies of this PDF, and repeatedly fill it out until you're good enough to do so in under 5 minutes, just as it says.

Sporcle offer this ingenious quiz for practicing radians and degrees. With help from the videos from the previous tab, this shouldn't be too tough.

The most complete quiz on the unit circle that I've found, however, is Math Fanatic's Unit Circle and Trigonometry Quiz. The menu lets you select exactly what aspects you want to practice, and the quiz itself lets you practice each section for as long as you need or want. Since it covers most of trigonometry, and not just the unit circle, there will be some quizzes not covered in my unit circle tutorial.

I hope you find this useful, and I also hope it helped you better understand the unit circle.

Memorizing Pi to 400 Decimal Places

The obvious first question is, “What exactly IS pi to 400 decimal places?” Here it is:

Pi=3.1415926535897932384626433832795028841971
        6939937510582097494459230781640628620899
        8628034825342117067982148086513282306647
        0938446095505822317253594081284811174502
        8410270193852110555964462294895493038196
        4428810975665933446128475648233786783165
        2712019091456485669234603486104543266482
        1339360726024914127372458700660631558817
        4881520920962829254091715364367892590360
        0113305305488204665213841469519415116094

There are many different ways to memorize pi. There are even those “pi purists” who refuse to use the mnemonic alphabet, and attempt to learn the numbers as numbers themselves. University of Edinburgh professor Alexander Craig Aitken learned it to a particular rhythm. Others assign meaning directly to the numbers themselves. For example look at the last four numbers in the first row above (1971). Some might remember this number as the year they were born or that some other memorable event from that year.

Are these methods effective? They certainly seem to be for the individuals who create them. It can be tricky though, for others to try and learn these methods, especially as the associations are often highly personal.

What advantages does this method offer? First, it can be taught to anyone who is familiar with both the mnemonic alphabet and the English language (indeed, it can be easily adapted to almost any Germanic language). Second, it doesn't just teach the digits in order, but out of order at the same time! What do I mean by out of order?

Imagine not just knowing the digits of pi themselves, but also where they are relative to each other. You could face challenges like these:

* Given the proper location, you can recall a corresponding group of four digits.
* Recall a single digit in the Nth position after the decimal point
* Given a group of four numbers, you can recall the location
* You can even recall entire sequences of numbers from pi

The traditional mnemonic alphabet method for pi is based on converting the numbers into words, and then linking them into a story. As you can see, if you forget just one element of the story, the entire number is thrown off! The method taught here eliminates the story aspect, and makes the memorization both simpler AND much more effective at the same time!

Prerequisites:

Link System
Major System

When you're ready, click to continue.