In this series of videos I'm going to teach you some techniques that I learned
from Dr. Norm Hoffman in his after-school mathematical problem-
solving classes, and I haven't seen these techniques taught anywhere else so these
were like my secret weapon that I used to get top marks in high school and
university. So nowadays I'm a maths tutor and I see a lot of my students
just memorising all of this stuff. Now you might have seen this before; you
might be completely new to trigonometry, which is fine, but in this
series of videos I'm going to show you how to work out all of this just using
the unit circle to its full potential. So I'm going to show you how to work out
these identities. I'm going to show you how to get all of this stuff straight in
your mind because it can be really confusing with... is sin 30 degrees a
half? Is it root three on two? If you're doing sin 150 for example, is that
negative, is it positive? You can work all of that out just using this diagram to
its full potential. And I'm gonna show you how to solve trig equations so
that's where this... these techniques really shine because they make it so
intuitive. I mean you can work out all this stuff using this ASTC mnemonic that
they seem to teach these days but it's just not so easy. Once you see how to use
this you're going to see how much better it is to use these techniques. So in this
series I'm going to cover all that stuff but for this video I just want to focus
on getting you used to the new definitions of sin, cos, and tan and how
we measure angles. Now I'm going to start with a question for those of you who are
already familiar with sine, cos, and tan, like maybe you've learned sohcahtoa or
something like that. So I want to know if we have a point
that's distance of one from the origin and it's at an angle of 40 degrees like
this, what is its x-coordinate, what's its y-coordinate, and what's its slope - the
slope of this line? So you might like to pause the video here and think about the
answers to those questions. Okay how did you go? To work this out what I would do
is draw in a little triangle like this, so the x-coordinate at this
point is just this distance here - let's call that x, and the y coordinate of...
of that point is that distance there. Now if you're familiar with sohcahtoa or
something like that you would know that cos of the 40 degrees there is adjacent
over hypotenuse so that's equal to x over 1 which is just x. So the x
coordinate is cos 40 degrees. If we do sine 40 degrees, that's opposite over
hypotenuse so y over 1 which just equals y. So the y coordinate is sin 40. If
we do tan 40, that would be opposite over adjacent which is actually rise over run
which is the slope. So tan 40 is the slope. Now... so we've got cos 40 is the x
coordinate, sin 40 is the y-coordinate and tan is the slope. This is
going to form our new definitions for sine, cos, and tan. Where sine, cos, and tan
previously were only defined for acute angles, now we can use these new
definitions for all angles whether they're obtuse or reflex or negative or
whatever. So I want you to memorise these new definitions: cos is the x coordinate,
sine is the y coordinate, tan is the slope. And this is just for
points that are a distance of one from the origin and where the angles are
measured in a certain way which I'm going to explain in a minute. So let's
suppose we're interested in estimating the sine, cos, and tan of 160 degrees. So
first of all we have to work out what does that mean by 160 degrees; where is
that going to go in the diagram? Well in mathematics the convention is
that angles are measured from here around in this direction. So you can think of
the 160 as being 90 plus 70, or you can think of it as
being 180 minus 20. Either way you're going to end up over here. Now this might
be confusing for some of you who have studied bearings before, because when
we're doing bearings we start from north and go in this direction so 160 degrees
would be down here if you're talking about bearings, but in mathematics the
convention is always that we start on the positive x-axis and go around this
way. Now when we're trying to visualise these trigonometric functions we always
want to keep this distance to be 1 so we're always going to be looking at
points that are on this circle with a radius of 1. So let's draw that in and
I've put a grid on here for you so that you can estimate the coordinates of this
point a bit better. So for now, see if you can figure out the sine and cosine of
this angle. You might like to pause the video here so you give yourself time to
think. Okay did you remember the new definitions that I gave you for sine and
cos? I said before that cos is the x-coordinate and sine is the
y-coordinate. So we can read the x-coordinate of this point
off the diagram: if this is at negative 1 the x-coordinate here is about -0.9
so cos of 160 is approximately negative 0.9. Now the negative is important so
make sure you've got that. For the sin, that's the y-coordinate of this point,
which you can see is at about 0.3 so sine of 160 is 0.3, approximately, and
it's positive. Now to work out the tan of 160 we may have to be a little bit
creative because it's not so easy to read the slope of this line off the
diagram. It'd be good if we had a really nice way to visualise what tan of 160 is.
Now because slope is rise over run, what we can do is make the run 1, and then the
slope will just equal the rise. So to make the run 1,
I'm just going to extend this line until it hits this line, and then this will be
our run of one. So now just focusing on this triangle, we have a run of one and
the rise: we can just read that off from the y coordinate here. So it looks like
about -0.4. So that's going to be tan of 160. Okay let's do
another one. Let's work out sine, cos, and tan of 220
degrees. So first thing we got to do is work out where is 220 on the diagram. You
might like to pause the video here if you want some time to think. So remember
that angles are always measured from here around that way, unless we're talking
about negative angles which I'll get to in a minute. So you can think of 220 as
180 plus 40 so that's going to end up around there.
Okay now let's work out sine and cos of that point. You might like to pause the
video here and see if you can figure it out. Okay did you remember our new
definitions of sine and cos? So remember cos is the x coordinate; sine is the
y-coordinate. So the x coordinate of this point looks to be around negative 0.8.
The negative is important so make sure that you got that. I don't really mind if
you said 0.7 instead; it's just an approximation, but the fact that it's
negative is quite important. So cos 220 is around -0.8. Sin 220
is the y-coordinate at this point which looks to be around -0.6,
again that negative is important so make sure that you said negative something.
Okay for tan I'm going to extend this line again until it hits this line like
that so see if you can work out tan of 220.
So pause the video here to give yourself some time to think. Okay how did you go?
Remember that tan is the slope, which if we've made the run into just one, the
slope's just going to equal the rise which is the y-coordinate at this point, which
looks to be about 0.8 so tan 220 is 0.8 approximately and it's positive. Now
let's try one with a negative angle. So what might negative 50 degrees mean? You
might like to pause the video here and see if you can figure that out. Okay
so if all the angles that we've been measuring so far have been going around
like that, a negative angle is going to be in the opposite direction so it's
going to be going around like this, so negative 50 is going to be around here,
like that. Okay so pause the video here and see if
you can estimate cos and sine of negative 50 degrees. All right, how did
you go? Remember cos is the x coordinate, so that looks to be around 0.6 and it's
positive. Sine is the y-coordinate which looks to be around negative 0.8, so make
sure you got a negative answer for the y-coordinate. Okay for the tan, I'm going
to extend this line again until it hits this line, and I've just had to extend
the grid a little bit as well so see if you can work out tan of negative 50. Pause
the video here to give yourself some time to think. Okay so the tan is the
y-coordinate at this point which looks to be around negative 1.2. Again that
negative is important so make sure that you got that right. Let's try one more:
380 degrees. Now where is that going to be on the diagram? You
might like to pause here to give yourself some time to think. Okay so if
we're starting here and going this way, 360 degrees would be here, and
then 380 is 20 more than that so actually past 360 up to here. So it's
actually just kind of the same as 20 degrees. Now what is the sine and cosine
of that angle? Pause the video here, see if you can figure it out. Okay
so cos is the x-coordinate which looks to be around 0.9 and it's positive. Sine
is the y-coordinate at this point which is approximately 0.3 and it's also
positive. For the tan I'm going to extend this line again like that so now see if
you can work out what tan 380 degrees is. Okay so remember tan is the y-coordinate
at this point which looks to be approximately 0.4 and it's positive. So
give yourself a pat on the back if you worked those out. If you didn't that's fine
you can just do some more practice. Ok so if you do want some more practice then
just grab yourself a calculator and just pick an angle at random, let's say 170
degrees. So just draw yourself a little diagram like that, it doesn't
have to be very accurate, we just want an approximation. Decide where 170 degrees
is on the diagram. So that one's almost 180 so it'd be like out like that.
Now should we start with cos? So what's cos of 170?
That's the x-coordinate of this point so just estimate... that looks like about
maybe negative 0.9 and then you can check the answer on the calculator by
typing in cos 170 and then you'll see... you'll see if you're right or not. It's
actually more close to negative 1. And then you could try estimating the sine
of that... maybe you think that looks like about 0.2 maybe. So let's check
that... yep about 0.2. And for the tan just extend that line... that was very wobbly,
doesn't have to be accurate... extend that line until it hits here and then the
y-coordinate here will be the tan. That looks like maybe negative 0.3. Let's test
that out... closer to negative 0.2. Okay but that's fine it doesn't have to be
very accurate. For the moment I just want you to understand whether they're
negative or positive and just roughly how big they are.
Okay so do some more of that practice if you need it otherwise in the next video
in this series I'll be covering how to derive these identities using the unit
circle so stay tuned for that one. And if you enjoyed this video then consider
subscribing and giving this video a thumbs up and I will see you in the next
one.