Review

For those who have come across this post by accident, this is Part 2 of a 2-part post on a tic-tac-toe-like game called Notakto. Part 1 can be found here, and will give you a more complete introduction to the game, as well as lessons on how to play and win on 1 or 2 boards against someone else.

Everything from this point on assumes that you've practiced all the strategies and can win 1- and 2-board Notakto games every time.

You should know and understand the importance of terms like sacrifice, boot trap, and 2X trap. You should also be very familiar with the few rules we've introduced so far:

• When playing on an odd number (1, 3, 5, etc.) of Notakto boards, you can guarantee yourself a win by being Player 1 - the odd-numbered player. When playing on an even number (2, 4, 6, etc.) of Notakto boards, you can guarantee yourself a win by being Player 2 - the even-numbered player.

• Your first move will always be placing your X in the center square of any empty board (empty board refers to any Notakto board with no Xs already on it).

• When the other player marks their X on a board with pre-existing Xs, your next move will be made on that same board. When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.

Starting from this point, I'll show you how to generalize what you already know to play and win Notakto on 3 boards, 4 boards, and beyond!

In the next section, you'll start by learning how to play 3 boards.

3 Boards

Let's start with the simplest example of a 3-board Notakto game.

Since there are an odd number of boards, you start the game, and mark an X in the center of any board. In this example, the other player responds by marking an X on the same board, and your reply is to sacrifice that board.


At that point, the game reduces to a 2-board Notakto game with the other player effectively going first. Even before they place their first X on the remaining boards, you should already know that you're going to win.

This is the basis of another general rule you should keep in mind:

Your basic strategy is to sacrifice boards, until you get down to the final remaining 2 boards. These remaining 2 boards will be played just as you would play any standard 2-board Notakto game.

Marking a Different Board

In this next example, you play first once again, but your opponent marks an X on the edge (it could just as easily be a corner or the center) of a different board. You recall this rule, and respond accordingly:

When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.

With 3 boards, all 3 now have an X on them. From there, you play to eliminate the first board you can, and play the rest as a standard 2-board Notakto game.


There are a few more lessons we can learn from this game. First, if your opponent doesn't mark a previously-empty board in the center, the defenses you've learned will usually prevent them from placing an X in the center of that same board until they're forced to. Of course, if they do mark a previously-empty board with an X in the center, you can easily sacrifice it or set up a boot trap, as needed.

Because of the rule that tells you when to be player 1 or player 2, combined with the rule that tells you when to place an X in the center of a previously-empty board, you can always guarantee yourself a minimum number of boards with Xs in the center.

With an even number, such as 4, going second and only marking previously-empty boards after your opponent does the same thing, it's easy to see a pattern of theirs-yours-theirs-yours guarantees you a minimum of 2 boards with Xs in the center (it's also possible ALL of them could have Xs in the center). Extending this to 6, you should easily see that the minimum number of boards with Xs in the center will be 3. Given any even number n, the minimum number of Xs in the center you'll have is n/2.

What about odd numbers? With an odd number of boards, you always start, effectively guaranteeing yourself an extra board with an X in the center. So, for any odd number of boards n, the minimum number of boards with Xs in the center works out to be (n + 1)/2. For 3 boards, that is (3+1)/2 = 4/2 = 2 boards minimum with Xs in the center. For 5 boards, you'll have a minimum of 3 boards with Xs in the center, and so on.

Why are the Xs in the center so important? When learning how to play 2-board Notakto, your defense depended on at least one of the two boards having an X in the center. Even if you didn't end the game on this particular board, it was the board with the X in the center which allowed you to keep control of both boards and win with either one.

Because of this, we're going to amend the rule taught from the simple 3-game demonstration above. The revised rule is below:

Your basic strategy is to sacrifice boards, until you get down to a set of boards, only 1 of which has an X in the center. On the sole remaining board with an X in the center, you will build your boot trap. If there are two or more boards without Xs in the center, play to sacrifice them until you get down to 2 boards (one of which is the board with an X in the center). These remaining 2 boards will be played just as you would play any standard 2-board Notakto game.

It's time once again to practice. Practice 3-board Notakto either online or with the iPad Notakto app until you can win every time.

Once you feel confident playing and winning 3-board Notakto every time, you're ready to apply what you know to win on 4 boards, 5 boards, and beyond! Just make sure you can win every time on a given number of boards, before moving on. You'll probably notice that it takes you less and less time to play perfect games on each level.

Once you can win every time on any level, it's time to stop playing against computers and learn how to present this as a game to real people.

How To Present Notakto

Now that you know how to play and win Notakto every time, it's time to learn how to present this to a real person. My first bit of advice comes from Bob Farmer, who would often include the following warning in his Flim-Flam! column:

Caveat Scamtor: Ethical Hustlers warn the Mark the game is fixed. Money lost is an educational investment. Gambling may be illegal where you live. Information in this column may be wrong, so don't bet the farm until you've verified it's right.

Since Notakto is so similar to tic-tac-toe, that's often the best way to introduce it. Here are a few key points that help when introducing the game:

• “Ever play tic-tac-toe? It's fun until you get to the point where both players are good enough that it's always a tie.” - This brings up the topic, and establishes familiarity.

• “My friends and I liked the fact you could play it almost anywhere, but hated those tie games, too.” - This takes the familiarity, and gives a reason for the non-standard rules you're about to introduce.

• “To prevent the ties, we decided that both players should play as X. After all, if you only mark Xs on the board, somebody has to get 3 in a row sooner or later, right?” - This introduces the all-Xs rule in a sensible way.

• “After some testing, we found it makes for a longer and more interesting game when the person who makes 3 Xs in a row is the loser instead of the winner.” - This brings up the other major rule change. Note that both new rules are introduces with the real reasons for their existence.

• “We've found it's even more fun when played on more than one board at the same time! When we do that, any board with 3 Xs in a row is out of play, and the last person to make 3 Xs in a row on the last available board is the loser.” - This quickly establishes the idea and rules of multiple board play. It is important for later than you do not mention a specific number of boards at this point.

Who Plays First?

At this point, you've already mentioned that the game will be played on multiple boards, but avoided mentioning exactly how many. This is about to give you the advantage in the game.

Ask the other player whether they'd like to go first.

If they decide that they're going first, draw 2 boards (or 4, if you prefer), and invite them to place their X anywhere. If they decide you're going first, draw 3 boards (5 is often too intimidating for a first game), and mark your X in the center square of any board, as usual.

You've subtly guaranteed yourself a win by having their choice of who goes first determine the number of boards!

If you want to state the number of boards before asking whether they go first, mention specifically that you'll be playing on 3 boards (or any odd number), and ask whether they'd like to go first. If they let you play first, play and win the game as normal.

If they decide they're going to play first instead, mention that, since the game is new to them, you should start with a practice game, so they can get the idea. Play the practice game, preferably playing to lose to build their confidence. Remind them once or twice through this game that it's a practice game. Once this “practice game” is over, play a legitimate game, mentioning that, because they went first last time, you get to go first this time.

You can find more tips on presenting Notakto in the next section.

Tips

• One more suggestion for the “practice game” technique: You could always have your practice game take place on a single board, explaining that this is to help them get the idea. Then, for the real game, you can move on to 3 boards as promised, in which you go first in that game, since they went first in the practice game.

• After you make each move during the game, you'll notice that usually all available boards will have an odd number (1, 3, or 5) of Xs. Naturally, this means that before you make your move, there will be only one board with an even number of Xs. This can act as a strong signal to tell you which board you need to play.

There are two exceptions to this pattern:

1) At the beginning of the game, when your opponent has just marked an X on a previously-empty board. In this situation, you're going to mark a new board with an X in the center square, of course. That makes this one case where each board played starts with an odd number of Xs before you play, while the same is true after you play (since you're marking a new board).

2) The other exception happens at the end of the game, the one described in the Attacked! section of this tutorial. If you're down to two boards, one of which has an X at the center and the other one doesn't, and your opponent sacrifices the board with the X at the center, then each move you make will leave an even number of Xs on the remaining board, instead.

• If you want to understand more about Notakto, here are a few helpful resources:

- Another puzzle (bgonline.org discussion, where the idea was first posted to the internet)
- Neutral Tic-Tac-Toe (MathOverflow discussion)
- Impartial Tic-Tac-Toe Presentation (PDF)
- The Secrets of Notakto: Winning at X-only Tic-Tac-Toe (PDF)


• This winning strategies don't need to be top secret. If someone genuinely shows an interest in learning how to play Notakto and win every time, feel free to teach them and/or send them to this tutorial. Above all, enjoy it and have fun!

Introduction

Math professor and Backgammon expert Bob Koca was playing tic-tac-toe with his 5-year-old nephew, when the nephew whimsically suggested that they should both play X. After mathematically analyzing such a game, Professor Koca realized that this was a deceptive new version of the classic game of Nim.

Thane Plambeck later dubbed this game Notakto (pronounced No-tac-toe). In this tutorial, you'll learn how to play this game so that you can win every time!

Notakto is played on a standard 3-by-3 tic-tac-toe board, and the rules are as follows:

• Both players alternate making an X on the board.

• Players may mark on X on an available space (again, any space not already occupied by an X) any available board during their turn.

• The person who makes a horizontal, vertical, or diagonal line of 3 Xs on the board is the loser.

If you'd like to try out this game for yourself, you can play the first level online here (click reset if you pass the first level to stay). If you have an iPad, you can download and play the Notakto app here.

When referring to general types of squares, I'll use the terms center, corner, and edge to apply to the various squares as follows:



When I need to refer to a square in more specific terms, I'll refer to the various squares with these terms:



I'll also frequently use the terms rotations and reflections.

If you rotate a given pattern of Xs through quarter turn (90°) increments to match another pattern of Xs, then those two patterns are rotations of each other. Two given patterns of Xs that are mirror images of each other, with left and right switched, and/or top and bottom switched, are reflections of each other.

These are important concepts, since patterns that are rotations or reflections will share the same strategy.

Now that you've got a basic understanding of the concepts involved, it's time to learn how to win 1-board Notakto!

Winning 1-board Notakto

Mathematician Timothy Chow originally likened the winning moves to that of a chess knight. If you're not familiar with chess, chess knights move either 1 square vertically and 2 squares horizontally or 1 square horizontally and 2 squares vertically, resulting in an unusual L-shaped move.

If we think of a chess knight on a Notakto board, you'll note that the knight in the corner square below (on your left) can only move into either of the edge squares marked with a dot. The knight located in the edge square (on your right) can only move into either of the corner squares marked with a dot.



Because of the unusual way the knight moves, a knight in the center doesn't have any possible moves.

Here's how to use the knowledge of a knight's move to win at Notakto.

To guarantee yourself a win on 1-board Notakto, you must play first, and start by placing your X in the center.

After each time the other player marks an X in a given square, you'll place your X a knight's move away from that square.

As seen in the knight's move graphic above, there will often be two open squares that qualify. Before placing your X, make sure that you're not inadvertently making 3 in a horizontal, vertical, or diagonal line of 3 Xs, thus losing the game.

Played properly, the resulting game should look something like this:

Why This Works

By starting with your X in the center, that limits every following move to be played on centers and edges, which are squares where you can use the knight's move strategy.

If the other player places an X in a corner at this point, the knight's move will place your X on an edge square. Conversely, If the other player places an X in the edge at this point, the knight's move will place your X on a corner square. This results in a boot-shaped arrangement of Xs at this point:


I refer to this arrangement (and any rotations and reflections of this arrangement) as the boot trap. Take a close look at it. The other player in the boot trap above won't place their X in the lower edge or the upper right corner, because they would instantly lose the game.

That leaves 4 remaining squares that seem harmless enough. However, when they place their X on any one of those squares, and you place your X a knight's move away (again, making sure you don't accidentally complete a horizontal, vertical, or diagonal line of 3 Xs), the board then looks something like this:


Notice that placing ANY X on this board must result in a losing play! From the boot trap, you can always force this situation, which means you'll always be able to win the game.

Practice 1-board Notakto online or with the iPad Notakto app until you can always win.

Once you feel comfortable enough with the 1-board strategy, it's time to learn to win 2-board Notakto!

Playing 2-board Notakto

For playing Notakto with 2 or more boards simultaneously, the rules are similar, but there are a few changes:

• Both players alternate making an X on the board.

• Once an individual board has a horizontal, vertical, or diagonal line of 3 Xs on it, that board becomes unavailable (no more moves may be made on it).

• Players may mark on X on an available space (any space not already occupied by an X on any available board) any available board during their turn.

• The person who makes a horizontal, vertical, or diagonal line of 3 Xs on the last available board is the loser.

While you'll still use the boot trap and the knight's move strategy, there are some new adjustments and new strategies to learn in order to win multi-board Notakto every time.

To guarantee yourself a win in 2-board Notakto, the other player must go first. In fact, there is a simple pattern that will help guarantee you a win with any number of boards:

When playing on an odd number (1, 3, 5, etc.) of Notakto boards, you can guarantee yourself a win by being Player 1 - the odd-numbered player. When playing on an even number (2, 4, 6, etc.) of Notakto boards, you can guarantee yourself a win by being Player 2 - the even-numbered player.

There's also a simple rule for remembering where to mark your first X:

Your first move will always be placing your X in the center square of any empty board (empty board refers to any Notakto board with no Xs already on it).

When you go first, all the boards are empty, so of course you can make your first move this way. If you're Player 2, there will always be at least one other empty board after the other player's first move, so you can be assured of still having an empty board on which to mark the center square.

These are great strategies for knowing how to start the game in your favor, but as I mentioned earlier, you'll need to learn some more strategies to assure yourself a win. The first new strategy is learning how to sacrifice a board.

Sacrificing

Since only the person who makes a horizontal, vertical, or diagonal line of 3 Xs on the final available board loses, taking other boards out of play by purposely completing a line of 3 Xs can be very helpful. Taking a board out of play in this way is called sacrificing, just like the same concept in chess.

Let's start with the simplest possible example of sacrificing a board. In the example below, the other player is Player 1, and you are Player 2 (because there's an even number of boards, remember?).

Their first move is the center square of one of the boards, and your reply is the center square of the remaining empty board. From this arrangement, no matter where they put their next X, there will always be 2 Xs in a line. Your response is to complete that line with a 3rd X, and sacrifice that board:


Once that sacrifice is made, the game effectively reduces to a 1-board Notakto in which the other person is the second player. As shown in the animation above, you set up your boot trap, and proceed to win the game just as before!

While the above game can and does happen, not every game happens in such a simple and straightforward manner.

Our next example will still show the concept of sacrificing a board, but it will happen later, and after a more complex series of moves.

This game starts, again, with the other player going first. This time, they're going to start in a corner (though it could just as easily be an edge), and you respond just as you should, marking an X in the center square of the remaining board.

This time, however, the other player marks an X on the same board on which you just played your center X. What do you do in this case?

Simple! You respond just as you would in 1-board Notakto, and set up your boot trap.

From here, we'll assume the other player marks an X in line with the X they played first. Naturally, you complete the row of 3 and sacrifice that board. The game then returns to the board on which you've set up your boot trap, and you win in the usual way:


This latter, more complex game actually shows a number of concepts that will be important through the rest of this tutorial:

1) Any strategy you learn in this tutorial can be delayed and still remain effective.

2) This is a another good lesson in determining which board to play:

When the other player marks their X on a board with pre-existing Xs, your next move will be made on that same board. When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.

In the previous tab, we discussed a similar rule that helps determine where you start. This rule, on the other hand, determines on which board you will continue play.

Try practicing 2-board Notakto online or with the iPad Notakto app using this approach, but don't be discouraged if you don't win.

Playing this way will guarantee a win if the computer responds as we've assumed above, but you'll notice that there are some situations that haven't been covered yet. The most common is when they start on a corner or an edge, and the next time they make a move on that board, there's no way to make 3 Xs and sacrifice the board!

In the next section, you'll learn how to use that kind of play to set up another trap for the other player!

The 2X Trap

When the other player marks their first X in a corner or an edge, it's quite possible that their next move could prevent you from sacrificing the board on your next move. It's not uncommon to see a situation such as this (the X in the right board is yours, the 2 on the left belong to the other player):


Another arrangement that could prevent an immediate sacrifice happens when two Xs are marked on edges that aren't directly across from each other (Again, the X in the right board is yours, the 2 on the left belong to the other player):


In both of the above cases, the next move is yours. What is the best move to make?

With both arrangements, the answer is exactly the same! You should place an X to make the pattern shown on the left board:


Why? The next time your opponent places an X anywhere on a left board with that arrangement, they'll wind up with 2 Xs in a line. When they do, you can place the 3rd X and sacrifice that board! Yet again, you can prepare your boot trap on the right board, and win the game as usual.

Because this design forces your opponent to place 2 Xs in a line, I refer to this as the 2X trap. It's easy to visualize, as the 2X trap always has 1 X in a corner, and 2 edge Xs, both of which are a knight's move away from the corner X.

Below is a full game animated, in which the 2X trap is used. This particular game involves a rotation of the 2X trap depicted above, so you can get used to seeing it in another arrangment. As with all 2-board Notakto games, the other player goes first.


As with the earlier example games, the 2X trap could be played even if you had developed the boot trap further on the left board. Delayed use of tactics, as I've said before, can still be effective.

Go practice 2-board Notakto either online or with the iPad Notakto app with your knowledge of the 2X trap, and you should find that you're winning more 2-board games than before.

However, your 2-board game still won't be perfect. Sometimes, your opponent can be sneaky and sacrifice your carefully-prepared boot trap! You'll learn how to handle that situation in the next section.

When They Sacrifice Your Boot Trap

Up to this point, every strategy discussed has involved sacrificing all but the last board (if any), and then winning the final board via the boot trap and the knight's move strategy that has been taught.

It isn't difficult to conceive of sacrificing a board, however, and your opponent can do it just as easily as you can. Below is a snapshot of a 2-board Notakto game where the following has occurred:

1) The other player moved first, and placed their X on an edge square (left board). You responded by placing your X in the center of the other (right) board.

2) They marked their next x on an edge square on the right board, and you responded preparing your boot trap.

3) They then marked an X on an edge square on the left board in such a way that you couldn't immediately sacrifice the board. You respond by setting up the 2X trap.

4) They then surprise you by sacrificing the right board, ruining your boot trap.

It's now your move. What do you do?


In this situation, there's only one effective response. You must place your X in the corner directly opposite the other corner X in the 2X trap:


Why does that work? While there are 5 remaining spaces open, the other player cannot place their X in the center or either of the open corner squares, because they'd lose the game. Their only possible response is to place their next X in one of the remaining edge squares, and your response will be to place your X in the other remaining edge square.

After they mark their edge square, and you mark your edge square, it's their move again, and the board now looks like this:


Anywhere they place their X on the left board, they must lose (Remember, the right board is out of play).

When They Sacrifice Early

Sometimes, when the other player sacrifices your boot trap early, the other board only has a single X on a corner or an edge (If the other board had a single X in the center, then you built your boot trap too early and on the wrong board), like this:


The response for this situation is easy, but different enough that it warrants its own section.

The strategy for this situation is simple. Your next move is to mirror the placement of the other player's Xs. What do I mean by mirroring the placement?

If they placed an X on the upper edge, you place your X on the lower edge (and vice-versa). If they placed an X on the right edge, you place your X on the left edge (and vice-versa).

Similarly with the corners, if they placed an X on the upper left corner, you place your X on the lower right corner (and vice-versa). If they placed an X on the upper right corner, you place your X on the lower left corner (and vice-versa).

In the case shown above, there an X in the lower left corner of the right board, and it's your move. The mirroring strategy tells you to place your X in the upper right corner, and keep mirroring their plays as discussed above.

If they mark a mixture of corner and edge squares, and you mirror those appropriately the boards will eventually look like this after your last move:


Does the pattern on the right board look familiar? It's the pattern of 6 Xs with an open diagonal (albeit a rotation of that earlier pattern) that ended the game discussed above.

Alternatively, if the other player keeps marking Xs only in corners, and you keep mirroring the appropriate corners in response, you'll come to this pattern much more quickly after your last move:


Even though there's only 4 Xs in this pattern, the other player must lose. It's their turn, and marking an X in the center or any edge means completing a horizontal, vertical, or diagonal row of 3 Xs, thus costing them the game.

With everything you've learned here, you're now equipped with enough knowledge to win every 1-board and 2-board Notakto game everytime.

To make sure you know how to use this knowledge, it's time again to practice 1-board and 2-board Notakto either online or with the iPad Notakto app.

Keep practicing until you can win the 1-board and 2-board games every time! Try and play without looking back at these strategies, but don't be afraid to look back at them when you need to do so.

Not only have you learned all the strategies you need to win every 1- and 2-board Notakto game, you've also learned all the strategies you need to play on any number of boards! In the next post, I'll take what you've already learned and show you how to generalize those strategies so you can apply them to any number of boards.

Rules

Somebody has mixed up the calendar! Can you help straighten it out?

Just like the 15 puzzle, the object of this puzzle is to return all the pieces to their correct order. Instead of 15 pieces to move around, however, the puzzle below has 41 pieces. In addition, the calendar aspect adds an extra dimension to the challenge!

Note the year and month above the calendar puzzle itself. Your challenge is to arrange the pieces so that they form the correct calendar for that month and year. The following rules apply:

• The blue square marked with a “1” will always be located somewhere in the top row. This should make sense. You don't start any calendar on the 2nd week.

• The blue numbered tiles are arranged on their corresponding days in the month. If the 1st fell on a Tuesday in the given month, then the blue “1” needs to be placed in the Tuesday column, and so on.

• The blue numbered tiles must be arranged in numerical order, reading from left to right, then top to bottom.

• The white lettered tiles are used to fill in the remaining spaces not used by the dates, and must be arrange in alphabetical order, reading from left to right, then top to bottom.

For example, if you're given a 30-day month that begins on Tuesday, the goal would be to arrange the puzzle like this:

Note, in this example, that A and B are used to fill the first Sunday and Monday respectively, since the month begins on a Tuesday. After the month ends, the remaining lettered pieces, C through K, are used to fill the remaining spaces.

The Year Range selection menus can be used to choose the range of years for your next puzzle (the default range is 2000 to 2099), and the New Puzzle button will generate a new month and year, as well as shuffle the pieces.

For help in solving the puzzle, try the next section.

Post your record times and moves, along with the year and month you solved, in the comments!

Solving

The two challenges that were combined to make the sliding calendar puzzle are taught separately here in the Grey Matters Mental Gym.

First, learn how to solve the classic versions of the 15 puzzle in this tutorial.

After that, you need to learn how to determine the day of the week for any given date.

To apply them to this particular puzzle, consider the year and month you're given, along with the first of the month. Once you figure out on which day of the week the first day falls, you also know which lettered tiles (if any) come before it.

Once you get the first lettered and numbered tiles in place, it's mostly like solving the original 15 puzzle.

When you get to the last two rows, the tricky part becomes working out which letter goes below which other piece. If you take into account where the letters in the top row (again, if any) left off, and how many days are in the month, this isn't too difficult.

Yes, you could cheat and look up the given month calendar online, but it's far more impressive to do it without looking.

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?