Happy Pi Day 2012

Happy Pi Day 2012 everyone!

For those of you who don't know what I'm referring to, today is March 14. Or, in one particular date notation, it is 3-14! If you want to be even more precise, you could have REALLY celebrated early this morning, at 3-14 1:59:26!

In honor of the annual Pi Day in mathematics, I thought that I would dedicate a brief post to one of the most common irrational constants in all of mathematics: π!

π is a ratio between a circle's circumference and diameter.  It is a constant for all circles, regardless of their size.    You are probably familiar with these common π-containing equations:

C = πd
C = 2πr
A = πr²

It is an irrational number, meaning that it cannot be expressed by dividing one integer by another integer.  So, π cannot be expressed as a fraction.  As a result, many aficionados in mathematics pride themselves on being able to memorize π to many, many decimal places.  In fact, π has been calculated up to more than a trillion decimal places!  You can view the first million decimal places at websites such as piday.org, though since it is irrational, the number of decimals continues infinitely!

In a fun website I just found, I recommend you give a read to 50 Interesting Facts about pi.  If you would like to learn more about the history of π and how it was first estimated, the wikipedia page for π also has a lot of historical information, from early methods of estimation by Archimedes all the way through to present-day computer approaches to calculating everyone's favorite constant, π!

Pythagorean Identities

Pythagorean Identities in trigonometry will show up very frequently and can be very useful.  I will explain how Pythagorean Identities get their name, how you can derive them, and how you can remember them.  First, it would be a good idea for you to be able to understand the basic trig functions sine, cosine, and tangent.  Once you are familiar with these trig equations, the algebra that we will apply to them will allow us to derive the Pythagorean Identities.  (If you would like a refresher, then I recommend visiting these links to have trigonometry sine explained, trigonometry cosine explained, and trigonometry tangent explained.)

The Pythagorean Identities get their name because they are based on the famous Theorem of Pythagoras.  You are very likely already familiar with it.  Simply, for a right angle triangle, it says "the square of the hypotenuse is the sum of the squares of the other two sides."  Mathematically, you have seen this represented as:

a² + b² = c², where a and b are sides and c is the hypotenuse.

Now, I will show you how to derive these special trig identities, using this theorem as our starting point.  To do this, we need to start with a right triangle, created by the radius of a unit circle and the axis:

How to label a right triangle inscribed inside a unit circle

We can say that the right triangle formed by dropping a line from the point that the radius touches the circle down to the axis has a base of x units long and y units high.  The radius in a unit circle, by definition, is 1.  Now, let's apply the definitions of sine and cosine to our triangle.  Recall:

sin(ɵ) = opposite / hypotenuse = y / 1 = y

cos(ɵ) = adjacent / hypotenuse = x / 1 = x

So, now we can relabel our diagram by substituting in these basic trig identities.

Right angle triangle with sides x and y renamed

With the triangle now correctly labeled for our derivation, we can apply the Theorem of Pythagoras to arrive at one of the Pythagorean Identities. Since a² + b² = c², we can therefore equate the sides of our triangle to these terms to give us our first of the trig Pythagorean Identities:

sin²(ɵ) + cos²(ɵ) = 1

If you have followed along up till now and understood everything I've done, then you are well on your way to remembering this trigonometric identity.  If you can remember how to derive you, you don't even have to memorize it (though it always helps!)  For the next Pythagorean Identity, you start with this first identity, and you apply some basic algebra and trigonometry to it to derive the second and third identities.  Recall the definitions of secant, cosecant, and cotangent:

sec(ɵ) = 1 / cos(ɵ)

csc(ɵ) = 1 / sin(ɵ)

cot(ɵ) = 1 / tan(ɵ) = cos(ɵ) / sin(ɵ)

With those inverse trig functions in mind, let's take the first Pythagorean Identity and divide all of its terms by cos²(ɵ).  That gives you:

1 / cos²(ɵ) = sin²(ɵ) / cos²(ɵ) + cos²(ɵ) / cos²(ɵ)

sec²(ɵ) = tan²(ɵ) + 1

And this is the second Pythagorean Identity!  Using the same strategy we just used to derive that one, go back to the first one and divide everything by sin²(ɵ), to arrive at the third Pythagorean Identity!

csc²(ɵ) = 1 + cot²(ɵ)

I hope that from this tutorial, you now understand how these identities get their name, how you can derive them, and how to use this knowledge to help you to memorize or recall them.  Using the fundamental trigonometry identities and trig relations, it is easy to come up with more advanced trigonometric formulas.  If you need to refer back to this Pythagorean Identities list, please bookmark this page and come back again.  Also, if you found this useful, please click the +1 button below.  Thanks.

Domain of a Line

When working with functions in mathematics, one problem you will frequently be asked is to find the domain of a line or identify domain and range of a graph.  This is a relatively simple problem to solve, once you know what is meant by domains and ranges.  If not, you might first ask "what is domain and range?"  I will begin with domain and range explained.

The domain is simply all the values of x that a function can take, whereas the range is all of the values of y that the expression can have.  A different way to think of this is to consider that domain is horizontally where the graph is located.  If you were to take a graph and squash it down to a single horizontal line, what values on the x-axis would it include?  Similarly, the range is vertically where the graph is located, and if you were to squish a graph from the left and the right into a single vertical line, what values on the y-axis would it include?

Another important problem you will have to work with is to determine if a given graph is a function or not.  This is also a much simpler question that many people first think.  The key to solving this type of question is to use the vertical line test.  If you have a function and you substitute in a value for x, you will only get one y value.  A function won't give you more than one value for y.  If it does, it isn't a function at all!  The vertical line test is a very quick and visual way of working with this principle.  To use it, you simply draw a vertical line through any point of your graph.  If the vertical line only passes through the line at a single point, then it is a function.  That is, the value of x corresponds to only a single value of y.  However, if the vertical line passes through more than one point (for example, a circle), then you do not have a function because for a value of x, there is more than one value of y.

So, now you know how to use the vertical line test to determine if your graph is a function or not.  Once you have decided that your graph is actually a function, you can then move on to finding the domain and range of a line.  Assuming that your line is plotted on a graph paper already with labeled points, finding the domain of a graph is incredibly easy.  All you have to do is identify the horizontal ends of the line, and say that the domain is between the left point and the right point.  The same strategy can be used to find the range of line graph.  An important note to make is that sometimes a domain or a range do not have end points, and so we say that they extend to infinity.

For example, if I asked you to find the domain of f(x) = x + 1, you can easily graph a straight line with a slope of 1, a y-intercept of 1, and the line extends forever up to the right, and down to the left.  This particular line has a domain of negative infinity to positive infinity.  So does the range.  However, if I just drew a straight horizontal line, starting at the y-axis and extending to the right to x = 5, then you would say that the domain is between 0 and 5.

Another point to make is about the notation used to describe the domain and range.  If the domain includes all of the numbers up to a point INCLUDING the point, then you use a square bracket [ or ] to represent that.  On the other hand, if the domain includes all values approaching a point, but NOT the actual point itself (you will find this situation more when working with limits and calculus), then you use round brackets ( or ).  So, for the previous question, I would record the domain as [0, 5] to indicate that the line extends from 0 to 5 and includes both of those points.  If it didn't include one, or either, of the points, I would use a round bracket next to the particular number.  On the graph, if the line only approached a point, this would be represented by an open point, like an 'o'.

When you advance to higher levels of graphing in algebra or calculus, finding the domain of a function will be a little more complicated, especially when you are looking at points that are irregular and cannot be precisely identified simply by looking at them.  Piecewise functions are also a concept that will really require you to understand domains.  However, I hope that this domain and range explained has given you at least a basic understanding of these very important math concepts, and with practice you will become an expert at finding the domain and range of a function.

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Best Measure of Central Tendency

What is the best measure of central tendency?  When working with statistics, you will find many situations where you need to choose the best measure of central tendency to most accurately represent your data.

The most common way of expressing central tendency is to report the average value, otherwise known as the "mean" value or arithmetic mean.  The mean is simply the sum of all the values in your data set, divided by the number of values in your data set.  You have undoubtedly used this calculation many, many times in your classes, perhaps without even realizing that it is called "mean" or that it is a measure of central tendency.

However, in addition to the mean, you can also describe the measure of central tendency by reporting the median value or the mode.  A median value is quite simply the middle value when you list all the values in your data set in numerical order.  For example, in the set 1, 1, 2, 3, 5, the median value is 2 (whereas the mean is 2.4).  Similarly, a mode value is simply the most common value in a data set (the mode is therefore 1 in the above set).  If there is an even number of terms in a data set, the set will have two mode values (the middle two).

So knowing how to calculate these different statistical averages, you can obviously see that each calculation results in a different value and so provides a different view of central tendency.  So, the obvious question then is "what is the best measure of central tendency?" or "what is the most appropriate measure of central tendency?"  Each situation will be different and so one method may be more appropriate than the others.

Keep in mind that you can correctly calculate each of the three measures, but reporting the most appropriate may sometimes be tricky.  Here are a few guidelines to follow when reporting these statistics.  (Remember, these are the best measures of reporting the "middleness.")

• If your data set is a normal and random distribution of data, (such as the heights of a group of people, or the temperature of a pot of water over a defined length of time), the mean value is most relevant.
• If your data set is skewed and contains outliers or extreme data points that would otherwise pull the mean too far in one direction (such as the job salaries of several workers, or test scores in a high school math class), the median value is most appropriate.
• If your data set is composed of non-numerical data (such as eye colour in a preschool class, or car type in a stadium parking garage), the mode will best tell you the most common data.

Consider these points when evaluating the best measure of central tendency for your statistical problem, and hopefully you will be able to most accurately represent your data.

Trigonometry - Special Angles

Trigonometry is the type of math that you use when you want to work with angles. Luckily, some angles are used so frequently that they have their own dedicated name and shortcuts that you can memorize. These are called special angles in trigonometry.

Special angles are great to know because their trigonometric functions equate to very specific and known ratios, so if you can memorize these it will save you a lot of time in doing trigonometry homework! To make things a bit easier, if you can't remember these exact values, it is even easier to memorize the triangles that these angles are based off of! And there are only two triangles, so you will find that it is very easy to derive the trig functions for the special angles if you can't remember them.

Specifically, the trig functions are easy to find for these special angles, which are: 0, 30, 45, 60, and 90 degrees.

This will hopefully make sense after looking at the triangles I mentioned.  Create a right angle triangle with two 45 degree angles, and with two sides of 1 unit length. By using the Theorem of Pythagoras, you can find that the hypotenuse of this triangle is easy to calculate to be length √2. This is what this triangle looks like:

So then, from these values and using the memorization trick of SOHCAHTOA, you can obtain the trigonometric values for this special angle of 45 degrees. You can work out that:

Sin(45) = 1/√2 

Cos(45) = 1/√2 

Tan(45) = 1

Don't worry if you can't remember these values and ratios. The easiest way to remember them is to memorize how to construct the special angle triangle. And as you can see, this triangle is very simple: a right angle triangle with a 45 degree angle and 2 sides of length 1, and you can easily fill in the rest and then work out the ratios yourself.

The second of the special angle triangles, which describes the remainder of the special angles, is slightly more complex, but not by much. Create a right angle triangle with angles of 30, 60, and 90 degrees. The lengths of the sides of this triangle are 1, 2, √3 (with 2 being the longest side, the hypotenuse. Make sure you don't put the √3 as the hypotenuse!). This triangle looks like this:

Here are the trig ratios that you can easily find:

Sin(30) = 1/2 

Cos(30) = √3/2 

Tan(30) = 1/√3

Sin(60) = √3/2 

Cos(60) = 1/2 

Tan(60) = √3/1 = √3 

Once again, just remember the triangle, and the ratios are easy to derive!

For 0 and 90 degrees, there isn't a triangle to remember (although please feel free to correct me if I am wrong!), so you will actually have to memorize these values. However, these aren't complex. I usually just remember the pattern of the following list:

Sin(0) = 0 

Cos(0) = 1 

Tan(0) = 0 

Sin(90) = 1 

Cos(90) = 0 

Tan(90) = undefined 

If you can't memorize the actual trigonometric ratios for the special angles, the key is to recall the triangles that describe them.  Make sure that you know how to construct the triangles, and then you can solve the trig ratios of the trigonometry special angles.  You will quickly find that doing trigonometry questions that use these special angles are easy!

Converting Slope-Intercept Form to Standard Form

In this post, I will explain to you how to go about converting slope-intercept form to standard form when working with an equation of a line.  Expressing an equation of a line in either slope-intercept form or in standard form will both describe the exact same line, despite the equations possibly looking different.  Furthermore, sometimes in your math problems you will be asked to provide your answer in one form or the other.  Therefore, it is important for you to know how to manipulate and rearrange the expression to convert slope-intercept form to standard form.

I will explain how to obtain the equation of a line in a different post, as well as a more detailed explanation of both slope-intercept form and standard form.

Slope-intercept form is the common way of expressing an equation of a line, and it takes the general form:

y = mx + b

You may be more familiar with seeing these equations expressed as point-slope form, which is closely related to slope-intercept form. In slope-intercept form, you can see your variables x and y, and the other two values (coefficents) are the slope (m) and the intercept (b). Point-slope form looks a bit different, requiring the slope (m) and a single ordered pair for a point on the line (x₁ and y₁), as well as the variables x and y:

(y - y₁) = m(x - x₁)

Plugging values into this equation, you will find it quite simple to rearrange it to obtain the slope-intercept form of the equation.

Standard form looks a bit different still, where the x and y variables are written on the same side, and A, B, and C are all coefficients:

Ax + By = C

The general strategy of converting slope-intercept form to standard form (or by extension, converting point-slope form to standard form) is to combine and simplify to rearrange the equation so that your x and y variables (with their coefficients A and B) are on one side, and the constant value (C, the terms without an x or a y in them) on the other side of the equals sign.

Here is an example of this type of question you might see.

Express y = 10x - 20 in standard form, and state the values for A, B, and C.

y = 10x - 20 (rearrange so x and y are on the same side)

10x - y = 20
A = 10, B = (-1), C = 20

It is important to note, as in this example, the value for B is negative! This is because the standard form has a PLUS in it. Pay close attention to the sign in your answer! So, more explicitly, the standard form of this equation could be seen as 10x + (-y) = 20.

Another point to consider is that during your simplification of the answer, you may end up with fraction coefficients. In this case, it is smart to multiply everything (both sides!) by the value in the denominator (to remove it from the denominator). Mathematically, if you are doing the same thing to both sides, you aren't really changing anything, which is what makes this allowed. Keeping all numbers as whole numbers and avoiding fractions where possible is a common practice.

If you want to go the other way, and convert from standard form to point-slope form or slope-intercept form, it is basically just the reverse of what we did here.  Try it with the above example to see for yourself.  Leave me a comment below if you would like additional examples or clarifications.  Please +1 this post below if it was helpful.

Quick poll!

Hi everyone and welcome again to my new website. So that I might be able to try and target better my visitors, I'd like to conduct a little poll. Could you please write me a comment below and leave what country you are from and what topics in mathematics you would be most interested in me posting about and explaining. Thanks for your help with this and I look forward to getting this site up and running soon with great content for you!