So now I'm going to give you some trig identities.
Some old ones that you may already remember and there are some new ones
which tell us how to add functions or add trigonometric identities
and then we'll also go over a small little proof that we will need to use
for the next proofs of sin, cos and tan. So let's start to look at our identities,
the first thing that you need to remember is sin squared x plus cos squared x equals to 1.
So let's just call this our first identity. So these are things that you need to familiarize yourselves
or know before we start to do these questions. Our second identity links tan and secs.
So we have 1 plus tan squared x equals to sec squared x.
Our third identity links cot and cosec, so we have 1 plus cot, squared x equals to cosec squared x.
And then we also said that I will tell you what cosec, sec and cot are.
So if we write them as identities here, cosec of x is basically the reciprocal of 1 over sin x.
So if you're ever face with 1 over sin, you can just simple write it as one 1 trig identity
and that is just cosec of x. Number five, we have 1 over cos x which is sec x or secant of x.
So sec x and lastly we have 1 over tan x which is cot of x.
Now, in easier way of remembering this is just to look at the third letter,
so if we have cosec of x so the s corresponds to the sin. If you look at sec,
the c, the third letter here corresponds to the cos and then if you look at the cot t,
it corresponds at the tan. There's a lot to learn here obviously
so sometimes we have to find easier or faster ways of working this out.
Now the next thing is, addition laws, so how would you add two angles within a sin, cos or a tan.
Now these are just rules that we can take on and just learn.
If I put them here so if we call them, just the addition laws,
they're also known as the double angle laws so you can derive the double angle formulas
from here which basically helps us do things like this.
So if you're adding sin of an angle, plus another angle
and these are fairly useful not just in the proofs that we're about to do
but if you have overlapping waves, so if you're doing something in medicine,
where you're measuring more than one wave, perhaps
and they're out of sync or at different angles. You can use these identities
to bring them together or combine them into one identity.
So sin A plus B gives us sin A cos B plus sin B cos A. This is our first addition law,
you can also subtract this but the sin would change so if you were looking at subtracting this,
you could, just put a minus here. So sin A minus B, would give you sin A cos B minus sin B cos A.
You can predict now, that we'll be doing exactly the same for cos and tan,
so the second rule here if we look at cos of A plus B. This is the same as cos A, cos B.
This time minus sin A, sin B. If you want to subtract it which we won't be doing in our course
but just so you know, in case you'd like to at some point, this is cos A minus B
and the sin changes here so that becomes cos A, cos B plus sin A, sin B.
Lastly, we'll look at the addition rule for tan, so we have tan of A plus B,
this is the same as tan A plus tan B over 1 minus tan A, tan B.
And again we will just be using the addition when we proof.
But if you were ever looking at subtracting two angles, you'd have tan A minus B,
this becomes a minus and that becomes a plus.
There's one more thing that I need to tell you before we move on to the next proof.
Now this isn't part of the identities and you won't really need to prove any point
unless you're doing advance mathematics and that is proving the limit.
So if we just put that here as well, that is proving that the limit of sin x over x
as the limit of x tends to zero is equal to 1. So you can imagine this,
we can put this in a different color here just so that you know that this isn't really
part of the course but we need this in order to prove the next identity of sin, cos and tan.
Now, it's very simple explanation, there's lots and lots of proofs for this,
the easiest possible proof here is to treat this as a circle.
So if I just go over for a half a circle here and imagine that this circle is one unit length
so we say that this length here is one and this length here is one,
essentially it's the ratio, we call this angle x and radiance.
This little bit here is the ratio of the arc length which is this length here
and the height of this triangle. If we remind ourselves of what the arc length here,
so the arc length of any circle is just R theta and this should be in radiance.
R in our case is just one and theta is x so the arc length here is just going to be x.
If we compare it with this length here, so just the tip of the,
as it touches the circle under the length down, so creating a right angle, triangle.
You can actually use one of the trick identities, so you can say that sin theta is opposite over hypotenuse.
Sin theta will just be sin x, the hypotenuse is just one so you can multiply this with one
gives you the opposite length. So what we really trying to say here is that this length here,
we can change that, the opposite length here is sin of x.
And the idea behind it here is just calculating the ratio of sin x over x.
So remember sin x in our case is this height, that height there that we're looking for
of the triangle and the x is the arc length there. So those are the two values that we are comparing.
So we're doing sin x over x and the idea behind it is, is that as we make this angle smaller,
so as we make this angle smaller and smaller and smaller.
They become so close to each other, almost equal.
So these values tend towards 1 as the limit of x tends to zero.
You can try this out with the few angles. You can try and put pi over 2 in here divided by pi over 2
and you can see what value you get, you can then out a smaller angle in,
so you can try and put pi over 4 into x and pi over 4 into x
and you will see that your answer is going closer to closer to 1.
And this is one of the easiest ways I can really prove this,
there are very long formal proofs to actually prove that sin x over x is 1
and as the limit of x tends to zero there is something called the squeeze theorem which helps you prove it.
So there are quite long derivative but graphically this is perhaps the easiest way to prove it.
So what I'm trying to say here is that as your x or the angle becomes small,
so as the limit of x tends to zero, we can say that sin x over x equals to 1,
tends to 1 or in this case we can just use through the definition equals to 1.
And again I'm only telling you this because we'll need this fact in one of the, in the sin and the cos proof.