Pythagorean Identities

Starting with the fundamental trig identities that I've already covered, a little bit of derivation leads to several new identities that will always hold true, and are especially useful in simplifying more elaborate equations.  Expanding upon my last post, there are a few other identities that can be derived from the Pythagorean Identity, which also go by the same name to describe the group of them, due to how they are deduced by making use of the his famous Theorem.  (Click here to get a great explanation about these Pythagorean Identities.)

If we start with the first Pythagorean identity, which I already explained:

Sin²(ɵ) + Cos²(ɵ) = 1

we can divide each side by the sine term to give something new.

[Sin²(ɵ) + Cos²(ɵ)  ] / Sin²(ɵ)  = 1 / Sin²(ɵ)

The first term (sin / sin) reduces to 1:

1 + Cos²(ɵ)  / Sin²(ɵ)  = 1 / Sin²(ɵ)

The remaining terms can be now be simplified.  We can apply the laws of exponents to group the terms and express them in a slightly different way, which allows us to apply additional trig identity substitutions to express things in terms of inverse functions:

1 + [Cos(ɵ)  / Sin(ɵ)]²  = [1 / Sin(ɵ)]²

1 + Cot²(ɵ)  = Csc²(ɵ)

This is the second of the three Pythagorean identities.  Hopefully my explanation is clear enough to understand, and you can follow along with my derivation of it.  As I always say on this site, if you understand the derivation, then you don't have to worry about making mistakes in memorizing the actual formula.  This is a perfect example of that, because the starting point is very easy to remember, but even I sometimes will confuse myself if I try to write down these additional identities without thinking about how to get there.  Learning the derivations of identities, formulas, and equations will always help you!

As I said, there are three Pythagorean identities.  This third one is derived in a similar manner to what we have already done, using the same sort of logical steps but dividing instead by the cosine term (rather than the sine term).  I am not going to write down all of the steps here, but I highly recommend that you grab a pencil and paper and quickly work through it, following along with the process that I outlined above.  There are only a few steps to work through, and in the end, this is what you should end up with:

1 + Tan²(ɵ)  = Sec²(ɵ)

As it is with many identities and equations in mathematics, it doesn't take very many steps to make something look completely different, as I've shown here.  I hope that you've been able to follow along and that everything makes sense.  If the steps in the derivations confused you at all, then I suggest going back to my posts on the basic trig functions, and working with those until you are comfortable.  The steps I have applied in these transformations were very simple substitutions, and I think that the most confusing part knowing what to look for in the first place.  With practice, you will learn to see patterns and recognize where you can apply various rules to make things simpler.

There are still several other trigonometric identities which I will show you how to derive in coming posts, many of which require the same types of thinking.  Stay tuned for more tutorials that explain things like the sine law and cosine law!

Fundamental Trigonometric Identities

Trigonometric identities are specific equalities that express one trigonometric function in terms of other trig functions. These fundamental trigonometric identities (also sometimes called basic trig identities) are fairly straightforward, but they take some work to derive them. If you are comfortable with simple derivations, you shouldn't have any problems though. Personally, I find it easier to remember the basic set of trig identities, and derive the more complex ones from those, rather than trying to memorize all of them... although some people are more comfortable just to memorize them.

These fundamental trigonometric identities are traditionally visualized as a right angle triangle inscribed within a circle of radius r, with sides formed by length x, height y, and the radius as the hypotenuse, r:

The basic trig definitions and relationships can easily be observed.  I have covered these trigonometric functions in other blog posts, so I won't go into them any further here, other than to just remind you of the concept of SOHCAHTOA.  Refer to these specific posts for more information about sine, cosine, and tangent.

Sin(θ) = y/r..... opposite  /hypotenuse
Cos(θ) = x/r..... adjacent / hypotenuse
Tan(θ) = y/x..... opposite / adjacent

For the first identity that I will show you how to derive, we need to make use of the Theorem of Pythagoras (I would also here like to refer you to my post about the Theorem of Pythagoras, which itself is derived from the cosine law, for more information on this trigonometric math concept.)  If we now apply the Theorem of Pythagoras to this right angle triangle we have inscribed within a circle, we can see:

r² = x² + y²

Dividing everything by r² gives:

1 = (x²)/(r²) + (y²)/(r²)
1 = (x/r)² + (y/r)²

And then, substituting in the basic trig identities listed above, we get:

1 = [Cos(θ)]² + [Sin(θ)]²

And that is the first of the 8 fundamental trigonometric identities. Nothing to it. In fact, this is an important mathematical expression.  As such, it has a special name or designation given to it to indicate that it is important.  It's proper name is the Pythagorean Trigonometric Identity. I'll rewrite it below in proper notation to clean it up a bit.  Notice how you can write the 2 (for the squares) in two different ways.  Either as above, like [Cos(θ)]², where you put the trig function in brackets to square it (note that you don't square the angle!), or more commonly as below, with the 2 placed next to the trig function to indicate that it is being squared.

Similarly, we can derive another basic relationship from a standard trigonometric identity.  For this one, it starts with:

Tan(θ) = y/x

Now to substitute in some more expressions.  This one is a little more complicated than how we arrived at the Pythagorean Identity above, but follow along and you should hopefully be able to see what I do.  The next step then, is to substitute in the basic Sine and Cosine definitions from the list at the top of this page (isolated for x and y, respectively) to give:

Tan(θ) = (r x Sin(θ)) / (r x Cos(θ))
Tan(θ) = Sin(θ) / Cos(θ)

And that's it again. Like the Pythagorean Trigonometric Identity I described above, this one also has a specific name.  This time, this one is called the Ratio Identity:

Those are now two of the simplest basic trig identities from which most of the others can be derived.  As I mentioned at the top, I always find that it is easier to memorize HOW to derive these fundamental trigonometric identities, rather than memorizing them explicitly and then possibly getting confused.  You WILL memorize them eventually (which always happens when you work with them so much!), but when you are first trying to understand them, I believe that it is always better to understand what is going on instead of just memorizing the final solutions.  This approach is a very wise strategy to apply throughout your mathematics training!


*Update 03-April-2012:  I have recently come back to this topic of basic trig identities, and have started a completely new post called the Trig Identities Cheat Sheet.  Please visit that site for a more thorough explanation of some of these derivations and fundamental identities of trigonometric functions.  Also, please click the +1 button below if you found this helpful!  Thanks.  

Stretching Graphs and Compressing Graphs

If you understand how to shift a curve horizontally or vertically, stretching graphs or compressing them isn't much different. Once again, it's only a small modification to the equation that causes the stretch or compression.  (Please click the Facebook Like button at the top of this post, or hit the Google +1 button at the end if you find this post helpful!)

Stretching and compressing graphs vertically is determined by the coefficient in front of the x (or more specifically, in front of the other direct modifications to x).
Let's look at a basic example.... f(x) = x², a standard parabola.

Now, to vertically compress this curve, you put a 'fraction coefficient' in front of the x component of the graph. i.e. f(x) = (1/2)*x². This squashes the graph down by a factor of 2. Or, another way to look at it, every y value in this curve is 1/2 of the value in the starting curve. Plot your own points to convince yourself of this. Note that the curve crosses (-2,4) and (2,4) in the original curve, whereas the new one crosses at (-2,2) and (2,2).

Now, naturally, if you put a whole number coefficient in front of the x term, you will be stretching graphs. For example: f(x) = 2x² looks like this:

You can see that this has caused the parabola to stretch upwards. Note that it now crosses (1,2), not (1,1). Or once again, to look at it from a different angle, every y value is now twice the value as in the original graph.  Compared to the original version, this is a vertical stretch graph.

The only other thing that you should keep in mind is that the coefficient to stretch or compress the graph vertically MUST be in front of any brackets that might be surrounding x, and the coefficient will act on any horizontal translation component and the exponent. Convince yourself of this by looking at graphs such as:

f(x) = (x-3)²....... and f(x)=2(x-3)²
f(x) = (x+1)³...... and f(x)=1/2(x+1)³
f(x) = x + 5...........and f(x) = 4x + 5
f(x) = (x-1)⁵ + 7...... and f(x) = 4(x-1)⁵ + 7

So far, all that I have talked about explains the concept of vertical compression and stretching.  But, let's now consider how we would go about showing a horizontal compression or stretch.  If we want to squash a graph together along the x-axis, what would we do?  Well, for any value of x, we would want the graph to be at some fractional value of that x.  So mathematically, this means that we have to modify the x value.  To perform a horizontal compression or stretch on a graph, instead of solving your equation for f(x), you solve it for f(c*x) for stretching or f(x/c) for compressing, where c is the stretch factor.  The simplest way to consider this is that for every x you want to put into your equation, you must modify x before actually doing the substitution.  Let me show you.

Consider again the parabola f(x) = 2x².  If we want to start plotting this graph, we could start by building a table of values and solving for f(1), f(2), f(3), etc.  Doing the quick math, you can see that f(1) = 2.  Now let's say we want to horizontally compress this graph by a factor of 10.  In this case, we do the modification on x before subbing in, so we apply the f(x/c) for compressing and see that f(1) becomes f(1/10).  You then substitute 1/10 in for x, and solve.  You then come up with the ordered pair of (1, 1/50).  This is a ten times compression along the x-axis.  If you draw some of these out, you will easily see that squishing or stretching along x is different from squishing or stretching along y.

Continuing with this same example, say we want to graph a horizontal stretch of a factor of 4.  We then solve f(c*x), giving us f(4*x).  For x = 1, f(4*1) becomes f(4), which works out to y = 32.  Again, it is easy to see how much we have stretched the graph by this simple modification.

As you can see, stretching graphs (or compressing them), both vertically and horizontally, really isn't that difficult.  There are obviously a few things that you need to remember, especially the distinction between horizontal and vertical changes.  But these modifications are just an extension of what you already know, building on your knowledge of horizontal or vertical shifts. Keep practicing, and you'll get it in no time. :)  (Remember to please hit Facebook Like button back at the beginning, or the Google +1 button if you liked my post!)

Manipulating graphs

In the next few days, I'm going to begin posting on graph manipulations. For any graphical expression, small changes to the expression can result in a very predictable change in the position of the graph. For instance, the graph can be shifted along the x-axis (horizontally), or along the y-axis (vertically). Similarly, the graph can be stretched or compressed along either axis as well. This may sound confusing, but the alterations to the equations are simple and easily recognizable. I will provide examples and go into detail in the coming days.

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 at the end, please hit the +1 button as well.

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.

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.

Please remember to Like and +1 this post if you found it helpful!  Thanks!