Trigonometry – Secant, Cosecant, Cotangent

In addition to the three basic trig functions we’ve already looked at (Sine, Cosine, Tangent), there are three other related functions. These are Secant, Cosecant, and Cotangent. These functions have similar meanings as the first three, in that they represent the ratios of various side lengths of a right angle triangle, and can be used to find angles or unknown side lengths. I will not go into extensive detail on these functions, as they are less commonly required, but I will show you what they mean. Please remember to click on the Like button if this is helpful, and be sure to follow me with the buttons on the right side!

So far, with the help of SOHCAHTOA, we have seen that:

Sine = opposite / hypotenuse

Cosine = adjacent / hypotenuse

Tangent = opposite / adjacent

These new functions are related to the originals because they represent the inverse ratios. (Of course, inverse means you swap the top and the bottom…)

Cosecant = hypotenuse / opposite… (compare to Sine)

Secant = hypotenuse / adjacent……. (compare to Cosine)

Cotangent = adjacent / opposite…… (compare to Tangent)

Also, these functions can be abbreviated: Cosecant = Csc, Secant = Sec, Cotangent = Cot.

At the middle school and high school math level, you will rarely have a need to use these functions, but it is good for you to know what they are. However, most trig problems at this stage can easily be solved with the original three functions. Just in case, though, it’s always good to know all the trig functions: sine, cosine, tangent, secant, cosecant, cotangent. If you’d be interested in additional material, a good place to start would be these links that explain secant, cosecant, and cotangent. You can also search that site for many other math concept explanations (just remember to come back here!)

Thanks for reading this quick “cheat sheet” version of these additional trig functions. Please remember to Like this post if you found it helpful, and bookmark my site to come back again!

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Graphing: Standard Form of the Equation

Just a short explanation for what is meant by “standard form” of the equation of the line. We have been looking at line equations in the form of y=mx+b. However, you may be asked to express this in standard form, or as a standard form equation. Graphing standard form equations will give you the exact same line as graphing something expressed as y=mx+b… standard form is just a different way of displaying the equation.

The general notation for a standard form equation is Ax + By = C, where A, B, and C are coefficients, and the x and y are the same variables we’ve been looking at but in a different position from what we recognize.

To express in standard form, you simply just rearrange the y = mx + b form such that you have x and y on the same side, equal to a number. Let’s look at some examples:

Given that y = 3 x+ 5, standard form of this is 3x – y = (-5).

Given y = (1/2)x -15, standard form of this is (1/2)x – y = 15… also, if you don’t want to have any fractions in your answer, you can multiply everything by the number in the denominator, such that we now get x – 2y = 30. Both expressions mean the same thing and will produce the same line. (In fact, convince yourself that no matter what you do to the equation, so long as you do it to both sides, the line is the same. eg. Multiply it all by 100, you get 100x-200y=30000… looks different, but it’s not! Reduce it down and see for yourself!)

For graphing standard form equations, you still might want to go from standard form to the mx+b form, for which you may need to do a bit more math, but it’s still quite straight forward.

Given 5x – 15y = 10, you just have to rearrange things to get y by itself on one side:

(-15y) = (-5x) + 10

y = (1/3)x – (2/3)…

and then you can see it is a line with slope 1/3 and y-intercept (-2/3).

Both types of equations mean the same thing. They are just expressed differently, and y=mx+b gives immediate information about the line without having to do a lot of work. However, you should be able to use both forms interchangeably. Convince yourself that graphing standard form equations will give you the same line as graphing y=mx+b equations. They just look different because the numbers are rearranged. This should be obvious because if you start with a standard form equation, and convert it to y=mx+b and graph it, you have only rearranged things not added or removed anything. You do not have a new line.

Also, from these equations, you should be able to tell that whenever you have an equation with 2 variables (x and y), and there aren’t any exponents on either term, then you are dealing with a straight line. So while an equation in standard form may not immediately look like a straight line equation to you until it looks like y = mx + b, because it has an x and a y in it (without an exponent… exponents make the graph do cool things later), it is automatically a straight line!

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Which Measure of Central Tendency to Use? Mode, Mean, or Median?

A concept which may need a bit more explanation is: which average is appropriate for a given question? What is the best measure of central tendency? When would you use a median instead of a mean, or perhaps use a mode instead?

For any data set, you can perform the analysis to come up with a value for each average. However, here are a few basic guidelines to help you choose the most appropriate form of central tendency to describe your data. If this is helpful, it would be great if you could please hit the +1 button to share it!

1. For a normal, random distribution of data (evenly distributed), the mean is preferred.

2. For a skewed data set, a median is more appropriate than a mean. The skewed data set (ie. extreme data points) will cause the mean value to be much more extreme than the median, and therefore less central.

3. The mode can be used for non-numerical data. Eg. hair colour in a classroom.

Here are a few examples of where each would be appropriate:

Mean:

1) students’ heights in a classroom

2) temperature over a length of time

Median:

1) income of a group of people

2) test scores for a group of students

Mode:

1) finding the most common hair colour in a room

2) finding the most common car in a parking lot

Hopefully these guidelines will help you to determine which is the most appropriate measure of central tendency to report for your data set.

If this post was helpful, please don’t forget to Like this post below, follow me on Facebook or Twitter, and any questions or comments of course are welcome!

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Math in the Real World (Video)

In many of my math posts, I reiterate that mathematics is not just the manipulation of numbers on a piece of paper, but rather it has real world meanings. You don’t have to constantly be performing math everyday, but that doesn’t mean that the world around you isn’t constantly abiding by the laws of mathematics. In fact, I bet you’d be hard-pressed to name something that ISN’T following some math rule. Check out We Use Math to see how important math is in our world, and the various kinds of jobs that math enables.

In addition to this, I came across a very interesting video that demonstrates how common, everyday things, that you probably take for granted, are very mathematical in nature. In it, you will see video footage of various scenes, alongside some example math formulas and depictions. This is a pretty cool video, very well done, and directly makes the point that I always try to convey – Math in the real world is everywhere! Please take a look. It’s only about a minute long, and when you’re done, be sure to leave me a comment below.

http://vimeo.com/77330591

For another bit of amusement, take a look at the menu on this post. Can you figure out the missing price? You need math in the real world if you want it!

Thanks a lot for reading my post, and be sure to follow me on Twitter (@TheNumerist) and Facebook (www.facebook.com/thenumerist). I really appreciate your comments and feedback, so feel free to drop me a message in those places or in the comments below. If there’s anything I missed or you want explained better, I can help with that too.

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Trigonometry – Tangent Definition

Be be successful in math, and especially trigonometry, you really need to understand the basic trig functions. The first of these functions I presented to you was sine, and then I followed that up with a closer look at cosine. Today, I want to talk about the third of the three basic trig functions – the tangent function.

If you’ve been following along up to this point, the tangent definition probably doesn’t even need to be explained to you, since it is so similar to its sine and cosine brothers. I’m sure you have figured it out already! However, for those who came directly to this page without going through those lessons, I’m going to give the full explanation anyways.

The tangent trigonometry function’s definition is another simple one. The tangent of an angle in a right angle triangle is the ratio of its opposite side length divided by its adjacent side length. This is as easy as it gets! To calculate the tangent of the angle, divide one side length by the other side length, and you’ve got your answer! If you want to see a cool interactive tangent calculator, take a look at Math Open Reference to see that demonstration. (Make sure you read the rest of my post first, though!)

In more general terms, you will find this relationship expressed as follows: TanB = Opposite / Adjacent. Tangent is often shortened to just tan, and in this case, TanB is saying “tangent of angle B”. You could have TanA or TanZ, or even use Greek letters to designate the angle (commonly theta for an angle).

So, once again going back to our basic right angle triangle, recall the triangle naming system, where the angles are shown by capital letters, and the sides directly opposite them are the lower case letters (e.g. the vertical side here should be name “b”). More importantly, recall the relative naming system, where the side names depend on what angle you are talking about. They hypotenuse is always the longest side, directly opposite to the right angle. Then, you have the side names “adjacent” and “opposite” to apply to the sides as they relate to the angle in question. For example, for angle B, the vertical side (AC) can be called “b” or “opposite” (because it is opposite the angle), where as the bottom horizontal side (BC) will be called “a” or “adjacent” (relative to angle B). It is important to realize that these names are relative, be the opposite and adjacent sides to angle B are NOT the same as the opposite and adjacent sides to A. If you don’t pay attention, you will quickly make errors in your trig calculations. This is a very basic concept that is easy to mix up, so it pays to understand it.

So then, we have TanB = b/c. We also have Tan = Opposite / Adjacent, which is easily remembered by our SOHCAHTOA trick. The last part is the key for tangent: TOA means “Tangent is Opposite over Adjacent.”

Working with the tangent trigonometry function is just as easy as doing calculations with sine or cosine. All you need is the basic side lengths are angles to be able to work out the simple relationship. Combining the tangent function with sine and cosine functions, you now have a solid understanding of the basic trig functions that will allow you to figure out any missing side length in a triangle. Furthermore, if you specifically have all of the triangle sides and are looking to calculate the missing angles, you can use the inverse trig functions (usually combine the ‘shift’ key on your calculator with the trig buttons) to fill in those gaps. (In this case, finding the angle when you already know the opposite and adjacent side lengths, you can use arctan.)

UPDATED – I’ll maybe make a small note here to also say that you will often see the term “tangent” when working with circles as well. At first glance, you may think that tangent actually has two separate meanings now (as I mistakenly implied originally). Whereas it can be used to describe the ratio of opposite and adjacent sides in triangles (as I have demonstrated in this post), the tangent of a circle denotes a line that touches a circle at precisely a single point (i.e. it doesn’t intersect or cross the circle at more than one point). The tangent line of a circle always perpendicular to the radius of the circle where it touches. You can connect these two thoughts with a circle. First, draw a circle, and then draw a radius from its center to its circumference. Then draw a “tangent line” (right angle) from the point where the radius meets the circle, and then connect this line back to the center. You have now created a triangle consisting of a radius, the tangent line, and the adjoining hypotenuse. If you consider the angle the the hypotenuse makes with the radius, say A, then by the definition of the trigonometric function, you have TanA = opposite / adjacent. The “opposite” in this case, is what we have already identified as the tangent line to the circle. So, hopefully this demonstrates to you that while we may talk about a “tangent line” to a circle, it all really comes down to the basic trig identity again. (Thanks to Five Triangles for pointing out my error!)

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Person Curves

This post is just for fun, but it does go to show you that mathematics is more than just numbers and equations. You can actually do fun and cool things with it! What I’m about to show you probably won’t get you any bonus marks on a math test, nor will you likely be able to actually do this by hand, but it is definitely amazing what a bunch of math formulas can produce! By now, you know that every equation gives you a differently shaped curve on a graph. You probably know what kinds of equations will always give you a parabola, and you may know what kinds of equations will give you a curve with 7 different humps on it. But you have no idea what the graphs of some of these complex equations actually look like!

Believe it or not, equations have been devised whose curves on a graph actually show you portraits of famous people! It’s true! Check out the graphs of Steve Jobs and Abraham Lincoln, shown below!

If you want to explore these amazing math masterpieces, Wolfram Alpha has a collection of line art portraits that are created from very complicated parametric functions. I’m sure these are devised by a computer. I’d have a very hard time believing that someone came up with any of these formulas! I’m not even going to pretend to understand some of the functions that are used in the equations. Nevertheless, this is probably one of the coolest graphing examples of complex math expressions that you will ever see.

Mathematica Stack Exchange has a more in depth analysis of how these curves are made, if you want to know a bit more about them. Otherwise, check them out in the gallery at Wolfram Alpha! Which one is your favourite?

Steve Jobs curve

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Why is X the Unknown?

Why does the letter “x” represent the unknown?

Probably one of the most common variables you will ever run across in mathematics is “x.” You have undoubtedly seen it countless times, in math problems asking you to simply “solve for x,” or as one of several different letters in a multivariable equation. Using the letter x to denote unknowns has even gone beyond mathematics – just look at The X Files on TV, where x indicates mysterious and unknown FBI cases, or even to the historical example of naming the x-ray, where x indicated a radiation coming from an unknown source. By now, you probably take for granted that x is the go-to letter when you want to represent some unknown quantity. While any letter can be used, I’m sure that you will agree that x is by far the most common one used. Have you ever stopped to wonder why? Why is x the unknown?

Where did this convention come from to let some seemingly random letter become the single one letter most often used to represent the unknown. Interestingly, it has its roots in language. Specifically, it is related to phonetics used in translating from Arabic, and the lack of corresponding phonetic sounds in Spanish. (LiveScience has a brief article on this here, and ResearchGate has a small discussion going over here.)

It is very interesting, and I will let Terry Moore describe in far more detail in the TED video clip linked below. It’s only a few minutes long, and it builds up to a nice tongue-in-cheek joke at the very end – especially if you have any Spanish friends!

Give this video a watch, and find out why x is the unknown!

Thanks again, and cheers!

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Trigonometry – Cosine Definition

In my previous post, I introduced you to one of the most fundamental, basic concepts of trigonometry: the sine definition. I explained to you what exactly it is, and how it can be used. Similarly, there are a few other basic trigonometry functions that you need to know so that you can solve these types of problems. In this post, I will introduce you to a very similar and related trig function: the cosine definition.

The cosine definition is actually just as simple as the sine definition, though you need to keep straight which is which. Let me first show you what cosine means (the abbreviation for cosine is just “cos”), and then I will reiterate the trick so that you don’t confuse these similar math functions!

The cosine of an angle in a right angle triangle is quite simply “the ratio of the length of the side adjacent to your angle and the length of the hypotenuse.” That’s all there is to it: a ratio of two numbers. Of course, the trick is remember which two numbers are specific to cosine. The good thing is that in a triangle, there are only three sides to choose from. But more importantly, there is a very easy memory trick so that you will never mix the ratio up with the sine ratio.

As I mentioned in my last post, this great memory trick I’m talking about is to just remember this completely made-up, nonsense word: SOHCAHTOA.

(MathsRevision has a pretty good trigonometry page that explains this concept similarly, with a different example and a short video).

If you break the word into its three components, you will understand quickly why this word is so great. Remember how the word sounds, then just put a space after every three letters, which gives you three words made up of three letters each: SOH CAH TOA.

The beauty of this word is that it tells you all you need to know about the three basic trig angles. The first letter of each component refers to the trigonometric function, and the following two letters represent the names of the sides that are involved in that ratio.

The key to understanding this is to remember the triangle naming rules, and what is meant by “opposite” side, “adjacent” side, and “hypotenuse.”

So what does SOHCAHTOA tell you? It tells you that “Sine is Opposite over Hypotenuse” (see the first letters? S.O.H. = Sine Opposite Hypotenuse). Similarly, it tells you “Cosine is Adjacent over Hypotenuse.” I’ll leave the last part for my last post in this three part series, but I’m sure you can figure it out pretty easily now.

So, there you have it! Now you understand what is meant by the cosine definition, how it applies to your trigonometry homework, and how to use it. For more information, or some great examples to practice with, check out this post at mathisfun.com. Also, the Wolfram Alpha page for cosine is also very helpful.

As I’m sure you’ve figured out already, my next post will be on the third of the three basic trig functions. (The suspense must be killing you!)

Thanks again, and cheers!

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Trigonometry – Sine Definition

Solving trigonometry problems can be easy, but it first requires you to have a solid understanding of the basic trig functions. There are three of these functions, and the first one I will discuss is SINE. In subsequent posts, I will highlight the other two functions: cosine and tangent.

The sine definition relates an angle of a triangle to the ratio of its opposite side and the hypotenuse. So, if we look at angle B in the picture below, (and keeping in mind the notation for sides), we can see that SinB = b/c. (If you’re interested, there is a more in-depth discussion on naming of trigonometric functions over at Physics Forums. Oak Road Systems also has an extensive explanation of the trig functions, and specifically at the bottom of the page for “what is sine“.)

In more general terms, SinB = opposite/hypotenuse. We write “Sin” as the shorthand form of Sine. (Sounds like mathematicians are kind of lazy… since it appears that writing 4 letters was too much, they had to shorten it to 3!)

Similarly, SinA = a/c. Again, it is opposite/hypotenuse… remember how I said it was important to understand the RELATIVE notation scheme! If you need a refresher, take a look at my prior post that goes over triangle naming conventions.

SOHCAHTOA is the trig acronym that describes the three basic functions and their ratios. It may be easier to remember if you look at it with spaces: SOH CAH TOA. The one that we are interested in this post is the SOH term, which is short for “Sine is Opposite over Hypotenuse.” There are many other mnemonics that can be used to remember the order of these relationships, and to help make this a more fun post, I encourage everyone to leave their favourite memory trick for SOHCAHTOA in the comments below. There is no one right way to have soh cah toa explained. Whatever works for you to help you remember is all that is important.

Working with the Sine function is fairly straightforward, and usually just a matter of plugging the appropriate numbers into the ratio.

For example, you can imagine a triangle with known side lengths, and be asked to find the angles. This is a very common and simple question that you will get. In this case, you would say Sin(B) = opposite/hypotenuse (where you substitute in the known values for the sides.) This will give you Sin(B) = “some value.” And now, just as in working with addition or multiplication, when you want to solve for a specific variable, you have to isolate it… and to isolate it, you must do the same thing to both sides. Therefore, to get rid of the Sine, you must do ‘inverse sine‘ to each side (which is usually the same calculator button, but pushing SHIFT to access it). Then you’ll get B = inverse sine of (some value), which is your answer. Technically, you are taking the arcsine of the ratio.

Similarly, if you know a mixtures of some of the sides and some angles, you may be asked to find the unknowns. You can then say Sin(known angle) = opposite/hypotenuse (where one of these sides is known and the other unknown), and then just simply solve for the unknown side length.

Here’s a quick triangle example:

So from this triangle, using the sine definition, we can tell that:

SinA = (4/5)

SinA = 0.8 (now push inverse sin on your calculator…)

A = 53.13 degrees

Also:

SinB = (3/5)

SinB = 0.6

B = 36.87 degrees

A useful trick for quickly solving triangles is to understand that if you sum up the three angles, they will always total 180 degrees. So for this triangle, after solving angle A, you could subtract it and 90 from 180 to find B. Of course, this defeats the purpose of practicing Sine in this example…

On the other hand, if we already knew that angle B = 36.87 degrees, and we wanted the length of the unknown hypotenuse, then we do:

Sin(36.87 degrees) = 3/hypotenuse

0.6 = 3/hypotenuse

hypotenuse = 3/0.6 = 5

As you can see, there really isn’t anything complicated about performing these steps. For the most part, it’s simple arithmetic with a few things about triangles thrown in for good measure. I hope this explains the sine definition so that it is understandable, and I also hope that now that you have had sohcahtoa explained, that makes more sense too. As always, please don’t hesitate to comment if you’re unsure or if you would like additional help. I’ll discuss cosine and tangent in the next few posts.

Thanks again, and cheers!

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Inverse Trig Functions

Inverse trig functions are a core concept in trigonometry, and you’re probably already used to using them without even knowing what they are. Specifically, they are named arcsin, arccos, and arctan (or, their full names: arcsine, arccosine, arctangent). We have already discussed how to find things such as the sine of 25 degrees, or cosine of 71 degrees. However, we need to introduce a new math concept to figure out such problems as “what angle gives a sine value of 0.2?” This is where inverse trig functions come in. With these functions, you start with the trig ratio and finish with the angle. Think of their relationship to the standard basic trig functions as being comparable to that between multiplication and division. You do one operation, and then the opposite to go the other way and undo the first operation. One way takes the sine of an angle to get the ratio. The other way takes the arcsine of the ratio to get the angle. Take a look at Math Open Reference to get another way of explaining it.

You may be taught these in school slightly differently. Instead of using the names arcsine, arccosine, and arctangent (arcfunctions!), it is also very common to see these represented as the basic trig notation with a -1 exponent, like these:

arcsine = sin^-1

arccosine = cos^-1

arctangent = tan^-1

(Interestingly, SparkNotes indicates that these functions are typically written with a capital letter, as in Arcsin, though I don’t seem to see that convention commonly used elsewhere. Leave me a comment below if you know more about this.)

These are likely the symbols that you will see on your calculator. Typically, they are the shifted function of the regular sin, cos, and tan buttons. It is important to note the distinction between the basic trigonometric functions and these new inverse trig functions. When using the basic trig functions, the value you are obtaining is the ratio of the two relevant sides of the triangle for the given angle. When using the inverse trig functions, what you are solving for is the actual angle that produces the given ratio of sides. So, make sure you push the right button on the calculator! Also, it is important to realize that these -1 exponent notations are NOT actual exponents indicating to take 1/sine, etc. They are simply a notation to imply the inverse trig functions.

Now, follow along with an easy trick that I explain below to help you work with inverse trig functions, and then go check out Math Warehouse, where there is a really good explanation, as well as a bunch of problems and a video for working with inverse sine, cosine and tangent.

The most basic way of finding an inverse function, in general, is a trick I was taught long ago, where you take your given function, substitute y for f(x), switch the x and y, and then rearrange.

f(x) = x² + 5

y = x² + 5…. now switch x and y

x = y² +5…. and rearrange

y² = x – 5

y = sqrt(x-5)

And there you have determined the inverse function. This is the same strategy that is being applied when we are talking about inverse trig functions. However, having the inverse trig buttons on our calculators really take all of this extensive and possibly difficult rearranging and calculating out of the picture.

Here is a basic example of one of these inverse trigonometric functions. Hopefully you will see that they are extremely easy to work with.

Find the angle for the given trig ratio:

sin(θ) = 1 /√2

θ = sin^-1(1 /√2)

θ = 45°

Hopefully this introduction to the inverse trig functions has been useful for you. There is a lot more information about this math concept that is probably quite beyond the scope of what is necessary to actually solve most of the basic trigonometry questions you will find in your maths homework. I have found several good resources, if you would like to learn more about inverse trigonometric functions. They can be found on the ubc.ca math website, or on The Math Page (among many other great math resources out there). Please leave a comment below if you know of any other great sites that you would like to share with other readers. Also, once again, please be sure to follow me on Facebook/Twitter/Google+ if you’ve enjoyed this post!

(This is an old post from my previous math site, In Mathematics, copied here to consolidate all my math pages.)

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