Okay special angles. Now don't panic because I don't expect you to memorise
all of this. We'll be figuring out which one is which from the unit circle. For
the moment what I just want you to remember is: for sine and cos, the
possible values are 1/2, √2/2, and √3/2. And for tan, it's
1/√3, 1, and √3. Those are the possible values that we're going to be
looking for. Now if you haven't seen this before and you're curious about how we
know these things, I'll just briefly go over this, I won't go into too much
detail, but the ones for 30 and 60 degrees, you get them by making an
equilateral triangle, cutting it in half and then focusing on one of the right
angle triangles left. Just apply SOHCAHTOA and Pythagoras' theorem and you
will get these facts out of that. And for the 45 degrees, you get that from an
isosceles right angle triangle, so where that side's the same as that side,
again applying SOHCAHTOA and Pythagoras's theorem you can get those
facts out of it. So this is what you need to know: for sine and cos the possible
values are 1/2, √2/2, √3/2, and for tan it's 1/√3, 1, and √3. Now
1/2 is 0.5, and √2/2 is approximately 0.7, and √3/2 is approximately 0.9.
Now you don't need to memorise this; I just want you to get an idea that this
is small, medium, and large. Now for the tan values, just think of this as being like...
well, 2 squared is 4 so this is going to be a bit less than 2; it's like 1.7 but
all you really need to know is that this is bigger than 1. And since this is 1
over √3, this one is gonna be less than 1. So that's really all you need to
know: small, medium, large, this one's smaller than 1, this one's equal to 1,
this one's bigger than 1. Now the way that I remember
which one is which, like which one is cos 30, which one is cos 60,
et cetera, is by just picturing the unit circle. And I'm going to be using the
definitions of sine, cos, and tan that I defined in my previous video, so check
that out if you haven't watched that already. So for example if I want to know
cos of 30 degrees, that's this angle on the unit circle, I think: what is the x
coordinate of that point? And I can see that it's the largest of these three,
like it's almost 0.9, so cos of 30 is going to be the largest of the three
possibilities, which was √3/2.
Let's do one in radians, let's go sine pi/4. Now you can think of that as
180 degrees divided by four, so that would be 45, or you can even think of it
like pi is over here and then half of that would be there and then half again
would give you a quarter of that, so that would be that one. So what is the y
coordinate of that point? You can see that it's the medium one out of these
three, it's about 0.7, so that one's gonna
be the medium one of the three possibilities,
so that's √2/2. Let's say we want to work out cos pi/3. So you can
convert that into degrees, you can go okay that's 180 degrees divided by 3, so
that's 60, so that's that one. Or you can even like visualise pi is this much, and
then if I cut that into thirds, it's like that's 1/3 of it, so we're talking about
this point here. And cos is the x coordinate and you can see that that's
the smallest x coordinate out of these 3, that's the smallest one, so that's going
to be 1/2. Now you can also... if it makes it easier for you to remember that
it goes small, medium, large, you can think of this as being √1/2, so then
the pattern is: √1/2, √2/2, √3/2.
Don't write that in a test; you'd want to simplify that down to 1/2. Let me
do one more of these. Let's say we want to work out sine pi/6. Now you can do
that as like 180 degrees divided by 6 is 30 so we're looking for sine 30 degrees,
that's this angle here, and sine is the y-coordinate which you can see... the
y-coordinate at this point is the smallest of these three, so that's going
to be the 1/2. Now let's move on to tan. So let's say we want to work out tan
pi/6, and remember that's tan 30 degrees which is this angle here. And tan
is the y-coordinate where you extend this line until it hits this one so
we're interested in the y-coordinate here. You can see that it's less than 1, like 1
is up here, so that's going to be one of the three possibilities for tan that's
less than one. So that was 1/√3. If we want tan pi/4, which is tan of 45
degrees, that's going to be this one, and you can see the y-coordinate there is
one, exactly 1. If we want to do tan pi/3, that's the same as tan 60 degrees,
which is this angle, and you can see that's a very large y-coordinate here,
that's bigger than 1, so that's going to be the √3. So you really just need to
remember that these are the three possibilities for tan, for the special
angles, and this one's less than 1, this one's equal to 1, this one's bigger
than 1. Let's move on to angles outside of this quadrant. So, 2pi/3,
where is that in the diagram? You can think of that as pi/3 being 60
degrees, and then it's twice as much as that, so it's 120 which is here, or you
can even think of like pi is out to here and we want two thirds
of that, which like if you cut that into thirds, it's that bit there. So that's
that angle, and we're interested in cos, which is the x coordinate of that point,
and you can see that it's about negative 1/2, so out of the three possibilities
it's going to be negative one over two. Let's try sine of -pi/4.
So remember pi/4 is 45 degrees and if we're talking about a negative angle
then we're talking about going this way, so negative pi/4 is out here, and sine
is the y-coordinate of that angle, which out of the possible special angles in
this quadrant, like negative 30 would be there, negative 60 would be there, it's
the middle one, the medium value out of those 3, so that's going to be the
middle of the three possible values for sine, which were 1/2, √2/2, and
√3/2 so that one's going to be √2/2, the middle one, oh and it's
negative, because the y-coordinate here is negative, since we're below the x-axis.
So that's negative √2/2. What about tan of 5pi/6? I would think of the
5/6 as being 1 minus 1/6, so 5pi/6 is pi minus pi/6, or you can convert that
into degrees and think of it as 180 minus 30, so that's going to be right
there. And if we're interested in tan then we
want to extend that line until it hits this line, like that. So we're interested
in the y coordinate at this point; you can see that it's negative and
that its magnitude is less than 1, like here's -1 down here, so it's
gonna be negative and out of the three possible values for tan, it's the
smallest one, the one that's less than 1, which is 1/√3. Now we can also
think about the sine, cos, and tan of the 4 sort of extreme points, or you can
think of them as the four multiples of 90 degrees. So pi/2 is 90 so
we're thinking of negative 90. Remember negative angles go around this way, so
we're talking about that point there. And sine is the y-coordinate, so what's the
y-coordinate at that point? That would be negative one, so sine negative pi/2
equals negative one, exactly. Of course you won't always have a nice
computer-generated, neat diagram to work from so I thought I'd do some by
freehand drawing to show you how that would work. Let's say we want to work out
sine of 2pi/3. As usual, pause the video if you want
some time to think and see if you can work it out for yourself. So what I would
do is draw the unit circle, it doesn't have to be too accurate, and
2pi/3, where is that on the diagram? It's 2/3 of pi which is here, so
it's gonna be like that, or you can convert it to degrees which is 120, so
that's there. Sine is the y-coordinate at that point;
you can see that it's quite high, like it's almost up to one, so that's gonna be
the largest of the three possible values for sine, so that is going to equal
√3/2, and you can see that it's positive as
well. Let's say we want to work out tan of pi. Let's draw the unit circle. Where
is pi on that diagram? Over here. So we are interested in the slope of this line,
so you can see there's no rise to that line, it's all run, so the slope of that
is zero, so tan pi just equals zero. Let's say we want to
work out cos negative pi/2, that's the same thing as cos of negative 90
degrees, if you want to convert that into degrees. So let's draw a unit circle,
negative 90 is down that way, and then cos is the x-coordinate, so what's the
x-coordinate at that point? It is actually just zero, so cos negative pi/2
is 0. What if we want to work out sine of 7pi/6? Now I would think of
that as like 7/6 is 1 plus 1/6, so we're interested in an
angle of pi plus pi/6, with pi/6 being 30 degrees, if you want to
convert it into degrees, so pi plus a pi/6 is that angle out there. And
we're interested in the y-coordinate at that point; you can see that it's
negative because we're below the x-axis and it looks like about a negative a half,
so yeah, that is exactly negative 1/2. Let's find tan of negative pi/3,
which, if you want to think of it in terms of degrees, is tan of negative 60.
So where is that on the unit circle? That's gonna be roughly there, doesn't
have to be too accurate, just make it you know bigger than thirty and bigger than
forty five. So for tan, you want to draw in this line, and then extend this line
until it hits that, and we're interested in the y-coordinate at that point, which
you can see is like bigger than negative one, negative one being there. So I mean
like it's... it's negative... it's a large negative number. So remember the three
possibilities for tan: 1/√3, 1, and √3. Root 3 is the biggest one,
bigger than one, so that's going to be negative √3. And I just wanted to warn
you about ones like tan pi/2. So on the unit circle, pi/2 is the
same as 90 degrees so we're talking about that angle there, and if you try to work
out the tan of that, it's actually undefined, because we're talking about
the slope of that line and it's a vertical line so it actually doesn't
really have a defined slope. If you try to extend this line until it hits this
line it would never hit it so we just say that tan pi/2 is undefined.
I want to show you how to use the calculator to verify these things, in
case you want some more practice. So let's say we want to work out something
like cos 150 degrees. So we draw our little unit circle. 150 is over
here. You can see that the x coordinate there is negative, and it's quite large, like
it's almost at negative one, so we might guess that that's negative √3/2.
Now if you're not sure about that, and you want to verify it, what you can do
is go to your calculator, type in cos 150, and then that tells you negative 0.866.
Okay so if you are not sure whether that's √3/2 or not, you can
type in √3/2 and see... yeah it's 0.866, so that's correct. Now... so
that's in degrees. I've currently got my calculator in degrees mode. If you can
see that little D there, that means that it's in degrees. If you want to work in
radians, you can switch your calculator into radians mode. On this model I have
to hit the "SETUP" thing, so SHIFT - SETUP, and then 4 is radians, so 4, and
now there's a little R there, I don't know how well you can see that, but there you go.
So now it's in radians mode. If I wanted to do something like...
let's say I want tan of negative pi/6, so I draw my little unit circle,
I go pi/6 is 30 degrees, so we want something like that. Tan of that
means we extend that until it hits there. I can see that the y-coordinate there is
negative and it's quite small, like it's less than... well greater than negative one;
its magnitude is less than 1. So I would guess that that's negative
1/√ 3, so the smallest of the possible values for tan. So if I want to
check that on my calculator, I type in tan negative pi/6, and that is
negative 0.577. So is that the same as 1/√3? Let's find out. One over
root three... 0.577, so yeah that one is correct.
So you can do as many of those as you like, try to figure it out, check your
answer on your calculator. So the methods that I've shown you in this video may be
quite confusing at first but once you get used to them they really are the
easiest way to work out sine, cos, and tan of special angles. Now in the next video
I'm gonna be talking about solving trig equations, and this is where the
techniques that Dr. Hoffman taught me really excel, so definitely check that
one out. And if you like this video then be sure to give it a thumbs up and if
anything still confuses you then leave me a comment below.