Identities! That's been the crux of my studies of late and I'm sure this won't be the last time I visit them. I took a snapshot of my handy-dandy notebook to the left - that page is full of trig identities, and that's just a few of them. There are countless identities that can be derived, and that's pretty fascinating honestly!
An identity is a mathematical statement that is tautologically true. That is, it is true no matter what numerical value is assigned to it.
Here's an example of a simple one:
tanɵ = sinɵ/cosɵ.
Since we know from graphing some functions that the sine of an angle is equal to its y coordinate on the unit circle, and the cosine of an angle is equal to its x coordinate on the unit circle, we can then derive that our original formula tanɵ = opposite / adjacent can also be stated tanɵ = sinɵ/cosɵ, which is honestly a pretty elegant way to relating our three major trig functions to each other.
There are three other trig functions that I've been neglecting mentioning and I guess here is as good a place as any to introduce them. These are the inverse functions: cosecant, secant, and cotangent. They're just the multiplicative inverses of sine, cosine, and tangent - respectively. Thus:
cscɵ = 1/sinɵ
secɵ = 1/cosɵ
cotɵ = 1/tanɵ ... or ... cosɵ/sinɵ
See how that works? You just invert them. Also note that the prefix co- is present in the inverses of only the functions that don't already have one: sine & cosecant (abbreviated csc), cosine & secant (abbreviated sec), and of course tangent & cotangent (abbreviated cot). These are the most basic trig identities. These statements are true no matter what the value of ɵ is.
I'm not going to go through proofs for a bunch of identities here (not in this post anyway - I may do one or two later if I feel like it'll help my understanding) - there are resources that will help you with that if you want to pursue them. But I will give you another basic one, and one that pretty much blew my mind.
An identity is a mathematical statement that is tautologically true. That is, it is true no matter what numerical value is assigned to it.
Here's an example of a simple one:
tanɵ = sinɵ/cosɵ.
Since we know from graphing some functions that the sine of an angle is equal to its y coordinate on the unit circle, and the cosine of an angle is equal to its x coordinate on the unit circle, we can then derive that our original formula tanɵ = opposite / adjacent can also be stated tanɵ = sinɵ/cosɵ, which is honestly a pretty elegant way to relating our three major trig functions to each other.
There are three other trig functions that I've been neglecting mentioning and I guess here is as good a place as any to introduce them. These are the inverse functions: cosecant, secant, and cotangent. They're just the multiplicative inverses of sine, cosine, and tangent - respectively. Thus:
cscɵ = 1/sinɵ
secɵ = 1/cosɵ
cotɵ = 1/tanɵ ... or ... cosɵ/sinɵ
See how that works? You just invert them. Also note that the prefix co- is present in the inverses of only the functions that don't already have one: sine & cosecant (abbreviated csc), cosine & secant (abbreviated sec), and of course tangent & cotangent (abbreviated cot). These are the most basic trig identities. These statements are true no matter what the value of ɵ is.
I'm not going to go through proofs for a bunch of identities here (not in this post anyway - I may do one or two later if I feel like it'll help my understanding) - there are resources that will help you with that if you want to pursue them. But I will give you another basic one, and one that pretty much blew my mind.
The Pythagorean Identity
The name will probably be familiar to you, and for good reason. To the right you can see a lovely graph laying it out, but I'll take you through the proof. It's pretty elegant really, the way it plays out.
We start with the Pythagorean Theorem:
The name will probably be familiar to you, and for good reason. To the right you can see a lovely graph laying it out, but I'll take you through the proof. It's pretty elegant really, the way it plays out.
We start with the Pythagorean Theorem:
a2 + b2 = c2
Now, apply this to the values that we know from our studies already. Given the unit circle definition, we can say that that c=1 because c is the hypotenuse, right? And we can also substitute sinɵ and cosɵ for a and b. Which gives us this:
sin2ɵ + cos2ɵ = 12
Which is to say...
sin2ɵ + cos2ɵ = 1
Which is to say...
sin2ɵ + cos2ɵ = 1
And there you have it. The first Pythagorean identity. It doesn't look like much until you remember that this is always true, no matter what the value of ɵ is. It pretty much blew my mind. We can derive other Pythagorean identities from this first one. Let's begin by seeing what happens if we remove the square of the cosine. To do this, we divide both sides by cos^2 ɵ.
cos2ɵ + cos2ɵ = cos2
Some of those things we've seen before, in this very post. sinɵ/cosɵ is the same as tanɵ. It follows that if we square the whole thing, like we have here, we get tan^2(ɵ). The second item, cos^2(ɵ) / cos^2(ɵ) equates out to one (because any number divided by itself is one). And finally, 1/cosɵ is one of our new functions: secɵ. Again, if we square all of that as it is here, sec^2(ɵ). So:
tan2ɵ + 1 = sec2ɵ
Neat, huh? That's another identity. It's always true. We can keep deriving more identities this way by playing with the way the equation is put together. Identities are pretty brain-bending at first, which is part of the reason this post took so long to prepare - but once you wrap your brain around them, the relationships between the different parts of everything starts to take on a new light.
And finally, I'll leave you with the solution to last week's trivia.
And finally, I'll leave you with the solution to last week's trivia.
When last we met, I asked you why the tangent wave approaches infinity at certain points. Well, let's look at how we find the value of tanɵ again:
tanɵ = sinɵ/cosɵ.
Now remember, the sine of any angle is equal to its y coordinate on the unit circle. The cosine is equal to its x coordinate on the unit circle. Now look at where the wave approaches infinity, and find those spots on the unit circle:
tanɵ = sinɵ/cosɵ.
Now remember, the sine of any angle is equal to its y coordinate on the unit circle. The cosine is equal to its x coordinate on the unit circle. Now look at where the wave approaches infinity, and find those spots on the unit circle:
Remember we're dividing y by x to find the tangent - what's the x value of π/2 and 3π/2? In both cases, zero. And what happens when you divide by zero?
Illuminating, Isn't it?
To infinity and beyond! See you next time.
Illuminating, Isn't it?
To infinity and beyond! See you next time.
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