What is Trigonometry?

Try asking any number of young math students the question "what is trigonometry?"  Inevitably, you will learn that it is one of the most feared subjects in math that students have to learn.  However, solving these questions and equations is much easier than they often give it credit for.  (For some interesting background history and applications of this discipline in mathematics, wikipedia has a good article.)  On this page I'm going to go over some of the basics to get you started.  Some of my following posts will go into more details on the specific functions, so be sure to check those out as well.  And please remember to hit the Like button and/or +1 button if this helps you at all.

Trigonometry Basics

Trigonometry, which literally means "triangle measurement," deals with the relationships between the angles and sides of triangles.  For the purposes of explaining the trigonometry basics, this is going to specifically deal with triangles that have one 90 degree angle (right-angle triangles).  These particular kinds of triangles have important distinctions to point out regarding their nomenclature.  The longest side, which is always the side that is directly opposite to the right angle, is called the HYPOTENUSE.  There is a common triangle notation of naming the sides and angles, as explained below, for which I will refer to this diagram:

The first concept that is essential for you to really understand is how to name a triangle.  The standard convention of naming triangles is to name them by the letters of their 3 corners. So in this example, this is triangle ABC. Similarly, the sides are named by the corners at either end of the side, so here we have sides AB, BC, and AC. The angles may be named with a single letter, or designated by 3 letters representing the points that make up the angle, in order, with the corner point in the middle (ie. angle B = angle ABC, angle A = angle CAB, angle C = angle ACB).

An alternate way to look at triangles is to name the ANGLES with capital letters. Then, the side that is directly opposite the angle is given the same letter, but in lower case. Therefore, from the above example, side AC = side b, AB = c, and BC = a. Somewhat confusing, but not really... it makes sense. The keyword in this method of notation is OPPOSITE, which tells you that the angle and the side opposite to it are related.  If this doesn't quite make sense to you automatically, you can also consider that the side c is the only one that doesn't help make angle C.  Check it out and see for yourself that this concept applies to all of the angles in any triangle.  If you can understand this basic naming scheme used in triangle problems, solving trigonometry questions will be that much easier.

In working with triangle sides and angles, you will also need to be able to recognize their RELATIVE positions.  So, in addition to the hypotenuse, which can easily be identified, the other two sides can be labeled relative to the angles.  So, this means that for angle B, you could be interested to label its OPPOSITE side and ADJACENT side. The opposite side is easy to identify: it is the side that does not touch the corner you are looking at. The adjacent side equally easy, though care must be taken to not mistake it for the hypotenuse: it touches the corner, but is not the hypotenuse.  Also recognize that in a right angle triangle, the two acute angles will always be created by an adjacent side and the hypotenuse.

A solid understanding of these trigonometry basics, including triangle notations and the naming of angles and sides (in particular, the relative naming designations) is important before you can move on to confidently do calculations involving these measurements. These calculations in themselves, however, are very straightforward, and as will be explained in separate posts, may be figured out very easily by remembering just one word - SOHCAHTOA.  This is one of the most important key points there is to memorize on this topic!  Make sure you check out some of my other posts (specifically, my lessons about sine, cosine, and tangent, where you will find sohcahtoa explained simply).

I really hope that I'm doing a decent job at explaining trigonometry easily.  Please hit the Facebook Like or +1 buttons to recommend my post if you found any value in it, and leave me some feedback in the comments below - especially if I didn't answer "what is trigonometry" well enough for you.

Why does "m" represent slope?

Why is the symbol for slope "m"?

Few people know how to answer this question properly, which doesn't help when dealing with inquisitive students!  There actually isn't a definitive answer to this question, and scholars are still looking for its first use!  Unfortunately, that likely won't cure any lingering curiosity.  And, frustratingly for teacher, this bit of mathematics history won't actually help students solve their graphing problems any easier, though it probably won't stop them from asking about it!

Some records indicate that it first appeared in print in the mid-1800's, in a geometry paper by a British mathematician named Matthew O'Brien, though other records suggest it dates back even further to Italy in the mid-1700's by Sandro Caparrini.  There is also a suggestion that it originated in the US (interesting email transcript here).  One theory suggests that because "slope" used to be called "the modulus of slope," they shortened this to just "m", though it is difficult to find evidence to back this up.  Similarly, the m may come from the words for "mountain"... in Latin it is mons and in French it is montagne.  There also is no evidence to support the myth that the French word for "to climb," monter, provides the "m." To dispute these suggestions, however, is the notion that the famous French philosopher Rene Descartes never recorded slope as m.  One would think that if the origin was from a French word, a noteworthy French scholar would have used it!  An alternate theory, which seems rational to me, is that the mathematician M. Risi suggested that the early letters a, b, c, etc are used to represent constants, the later letters z, y, x, etc are used to represent unknowns, and the middle letters represent parameters.  Therefore, since slope is considered a parameter, it may have arbitrarily been assigned this mid-alphabet designation.  For all we know, it was a practical joke that got taken too far and is now an accepted concept of the mathematics of graphing!

Whatever the true origin of this seemingly strange symbol, it won't provide any insight into easier ways to work with slopes, lines, and graphing!  So, the next time someone asks this question, you don't have to feel bad to say "I don't know and I don't think anyone else does either.  Now go do your homework."  :)  If you enjoyed reading this short post, please do me a favor and click the +1 button below to share this wealth of information!

Functions - Complex, Piecewise

The last post introduced you to functions and how they work, but there are few other things worth discussing. The first is more about function notations.

When I introduced functions, I referred to the standard notation as f(x). However, you will quickly come across problems that use different letters, but mean the same things. You will probably see things like g(x) or h(x), and they are just used to differentiate between different functions. An example will probably clear this up:

f(x) = 2x + 3
g(x) = x - 8
h(x) = 4x + 1

That shows that function f (or g, or h) is equal to the given expression. So f(x), g(x), and h(x) all represent different functions. It's the equivalent of saying 3x = 12, or 5a = 10... the math is the same, just with different letters.

So, with multiple expressions, we can do slightly more complicated things. If we say solve f(1), we substitute 1 in for x, and solve it to get 5:

f(x) = 2x + 3
f(1) = 2(1) + 3
f(1) = 5

However, we can now ask for something like f(g(x))... read as "f of 'g of x'." Looks complicated, but keeping in mind what the function notation is saying, we can easily solve this by substituting the expression for g(x) in for x (instead of just subbing in a single number like before):

f(x) = 2x + 3, g(x) = x - 8
f(g(x)) = 2(x - 8) + 3
f(g(x)) = 2x - 16 + 3
f(g(x)) = 2x - 13

Furthermore, if we wanted to solve f(g(5)), we can do one of two things. We can evaluate the expression as we just did, then solve when x= 5... or we can solve g(5) first, then substitute that into f(g(x)):

Method 1:
f(g(x)) = 2x - 13
f(g(5)) = 2(5) - 13
f(g(5)) = (-3)

Method 2:
g(x) = x - 8
g(5) = 5 - 8
g(5) = (-3)...
then...
f(x) = 2x + 3
f(g(5)) = f(-3) = 2(-3) + 3
f(g(5)) = (-3)

Both methods get the same answer, because they are doing the same things. It's just a matter of what you are more comfortable with. Function notation can get very confusing like this, but for problems like this, you just have to basically work from the inside out, just as in any other math expression you've seen! (eg. (2x + 7(3x -4) + 2).... solve the inside bracket first, then the outside). They may seem tricky, but you will be surprised at how easy they are when you see through the notation and know what they are asking!

Another point to discuss is functions that are not described by a single equation, but remain continuous. Remember back to what we said a function is... a way of relating two variables, where each number in the domain corresponds to ONLY ONE number in the range. Also remember the vertical line test. But we never said anything about it being entirely expressed by one equation! So then, we can say that the following is also a function:

Obviously, it is composed of two straight lines, each having their own equation. This is called a PIECEWISE FUNCTION. Since it is just one line with two distinct 'pieces,' it is a single function, not two, and it is written as:

That says that f(x) is described by two separate equations, over the domains of x stated after each equation. Given this graph or equation, if you were then asked to solve f(2), you must look to see within which domain x = 2 falls, and evaluate that expression. For this example, f(2) is in the top equation, so f(2) = 3. You can check by looking at the graph to see that when x = 2, the function evaluates to (y equals) 3. Alternately, f(4) is within the bottom expression, so f(4) =3/2.

Try to remember that as difficult and tricky as some functions look, they aren't that difficult if you relate to them what you know already about solving math problems and you pay attention to what is being asked! Remember, if you need something clarified further, don't hesitate to post a comment!

Functions - Domain, Range, Vertical Line Test

Functions may seem like a totally foreign and incomprehensible concept at first, but they're not really the monsters that many students think they are. The notation may seem a little weird from what we're used to, making them look difficult, but when you understand the concepts of what is being presented, you will see that they're not really doing anything that doesn't make sense.

To give a very simple definition, a function is basically just a way of relating two variables... although their notation makes it look otherwise. But we've already seen something like this before! If we take a regular X-Y graph and draw a straight line on it, the equation of the line is the way of relating the x variable with the y variable. However, the standard way of writing a function is to write f(x) instead of y. We say it like f of x. So for example, taking a line, we can do the following:

y = 3x + 5
f(x) = 3x + 5

The function notation says that "the whole expression with x on the right" evaluates to some number (the equivalent of y). The math is still the same, no tricks with what you can do with the expression. It's just a different way of describing things. But we also now have a way of saying "tell me what that thing evaluates to when x=10" and we can write that out like this:

f(10) = 3(10) + 5 = 30 + 5 = 35
f(10) = 35

Again, it's essentially the same thing as saying 'solve for y' from our graph equations, but you can't say to solve y = 3x + 5 at a certain point for x, without actually writing that instruction out in words. With function notation, you can say to solve the expression when x=10, 50, or 25,000, just by writing f(10), f(50), or f(25,000).

There are a few other terms you will likely see when dealing with functions. Domain refers to all the values of x that are valid for the expression. Range refers to all the values of y in the expression. Domains and ranges could be infinite, as in the case of a straight line that goes forever in either direction, or they could have defined numbers as in line segments. The notation for domain and range sets is like [x₁, x₂] or [y₁, y₂], where those numbers represent the two extremes of the domain (furthest left and right) or range (highest and lowest), and the square brackets indicate that those numbers are included in the range. (If you had an 'open' point on a line, this indicates that that exact point is excluded from the line, and if this occurred as an endpoint, you would write (x1, x2) to denote that the number is not included in the set. Also, you could have on the line a closed and an open point, in which case you'd have something like (x1, x2] or vice versa.)

The domain (the x-values) for line segment A is [-8, 1] and its range (y-values) is [5, 8]. In this case the domain and range extremes are the actual end points of the line, but pay attention to the particular question as this is not always the case. Remember, this notation is defining the domain and range (all the values that x or y could take), not the end points of the line.

Similarly for line segment B, the domain is [2, 7] and the range is [-3, 6].

Line C demonstrates a very important concept about functions. Line C is not a function. We should expand our definition of a function to say that it is a way of relating two variables, where each number in the domain corresponds to ONLY ONE number in the range. That means, that for every x, there can only be one y. To test this on a graph, you can use the vertical line test to prove a line is a function or not. If you draw a vertical line through your function at any value for x, the vertical line will pass through the function line ONLY ONCE. If it passes through it more than once, then you are not looking at a graph of a function. Therefore, looking back to line c, you can tell that a vertical line would pass through the line twice at some positions, and so this proves it is not a function of x. Similarly, the circle in D does not pass the vertical line test, and so is not a function either.

These concepts may seem different, but you will become accustomed to them. Just remember our definition of what a function actually is, and what the notation is saying, and you will be fine. When you begin to grasp these concepts and advance to higher levels, you will get to see some really cool stuff with functions. (sneak peek.... f(x) = x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.... it's still a function!)

If any of that isn't clear and you'd like something more explained or clarified, please don't hesitate to leave a comment for me! I'm always glad to help!

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.  (Please hit the Like button and/or the +1 button if this post is helpful for you!)

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.

(check this post for some additional pointers)

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