BAM!!! Mr. Tarrou. Ok, we are going to revisit that unit circle and start evaluating some
trig functions that we can figure out. 16 of the basic trig functions...well there is
6 trig functions and 16 angles and use these to in cooperation to answer some questions,
to set up some trig values and keep them in exact form. We don't need to use a calculator
and most of your Precalculus and trig teachers should not be allowing you to use your calculators
on these questions. Because, you know, we don't want just a decimal you get punched
out of a calculator. That is not going to show us that you know this and understand
this unit circle. So any trig or precalculus question that is based off or uses one of
these 16 angles, or some half and double angles which we will learn those identities, but
right now we are just using these 16 angels, they should be left in exact form. That means
no decimal numbers like umm.. Like if the answer is square root of two, leave it as
square root of two. Don't make it 1.datta datta datta with all the decimals. We have
got... When you set up your unit circle don't forget that you count all the way around by
thirty degrees and then all the way around using 45 degrees. Or if you are doing radians,
count by pi over 6 and then pi over 4. That will give you all sixteen angles that you
need for the unit circle. Also remember that besides all the quadrantal angles that we
just identify easily as moving to the right or left one unit or up or down one unit from
the origin. We do not need a triangle, nor can we set up a triangle, for those four angles.
The only angles that we can set up triangles for are the 12 angles, each of which you find
three in each quadrant. There is only two triangles making up this entire unit circle.
The 60-30-90 triangle and the 45-45-90 triangle. That is why you see all these repeating coordinates
and why look so familiar. The only difference is, as we can see in these orange and yellow
one, in quadrant both the x and the y are both positive and in quadrant four where we
have rotated 330 degrees or 11pi/6 radians we are using the same 30-60 triangle. But,
since we are down in quadrant four the y value is negative and the thus the same triangle
can set up not just two but four different angles. Ok, let's move on to evaluating some
of these trig functions. The sine of theta is y over r. The cosine of theta is x over
r. The tangent of theta is y over x....and so on. I am sure you can read the rest of
these. Please note that in a trig function you put in an amount of rotation, either degrees
or radians, and you get out the sides of a triangle. Ok, so in an angle and out a triangle
with your six trig functions. Let's evaluate a couple. So let's say that we have the cosine
of pi over 3. We have to remember soon that pi over 3 is 60 degrees. You will also need
to remember that...sorry. I forgot one thing. We are talking about evaluating these trig
functions and these trig functions work out to be the same regardless of the size of the
circle, but I have set up a unit circle. When you set up a unit circle, your r values are
always going to be the value of one. That is the definition of a unit circle. And so
if we are holding the r value to be a value of one on the unit circle, the cosine is just
going to be the x coordinate and the sine is going to be the y. As such I have identified
that here in green. So the cosine of pi/3 or the cosine of 60 degrees is just the x
coordinate of one half. Let's go in the opposite direction. The cosine of negative pi over
three. Well positive rotations go in a counter-clockwise rotation from zero, the positive x axis. And
negative pi over three goes in a clock-wise direction. Negative pi over three is going
to land you down here at negative 60 degrees, or as you can see what I am pointing to negative
pi over three is the same as five pi over three or 300 degrees. Those are coterminal
angles, they are separated by 360 or 2pi. At any rate, it is still going to be the same
value at one half. So are you always going to get the same answer when you take a trig
function of an opposite angle like this. The answer is no. It is not going to work for
sine. The sine of pi over three is the y value at 60 degrees which is square root of three
over two. The sine of negative pi over three, again that clockwise rotation as opposed to
counter-clockwise rotation, is going to be negative square root of three over two. Right
there. Why, because sine is based off of the y coordinate and when you are down in quadrant
four, or quadrants three and four, your y values are going to be negative and thus your
sine is going to be negative. We will be learning later that the cosine function is an even
function. I can show you here. When you plug in opposite values of x you get the same value
of y. If you are not sure what even functions are you can search my channel, type even and
odd function into my search window or just even function or odd function, and you find
some videos for that. When you plug in negative or opposite values of x, you get the same
value for y. With the sine function, you plug in opposite values of x or theta and you get
out opposite values of y. I am not to say x and y in terms of the unit circle, I am
just saying in... I probably should be saying domain and range. You plug in opposite values
of the domain and you get the same value for the range. You plug in opposite for the domain
and you get opposite values for range. The is odd and this is an even function. Let's
take a look at tangent. The tangent of theta is y over x. Let's say...let's find the tangent
of 4pi over three. Ok, 4pi/3. That is one-third, two-thirds, three-thirds, four-thirds. Anything
with a divisor of three is a multiple of 60 degrees. So one sixty, two sixty, three sixty,
and four sixty. Tangent is y which is negative square root of three over two over x which
is negative one half. You guys...my students have seen me do this, when you have a fraction
over a fraction the bottom one is going to flip up. That is because you can't really
divide fractions, you have to multiply by the reciprocal of the second fraction. Well,
the second fraction is the denominator. So, negative square root of three over two divided
by negative one half becomes negative square root of three over two times negative two
over one. The two's cancel out, the negatives cancel out and you get a final answer of square
root of three. Now ok...that is good. Remember that I picked an angle that is in quadrant
three. Now tangent is y over x. In quadrant three both your x coordinate and you y coordinate
are going to be negative thus your tangent in quadrant three is going to be positive.
Just make sure you get that positive answer. Let's pick another angle measure. Let's find
the tangent of 270 degrees. I went from radians to degrees now for you. We are using all these
angles off the unit circle by the way, so we know if we are looking at degrees or radians.
If you are using your calculator, and it would not be for one of these 16 angles, but if
you are using your calculator the only time you have to worry about what mode your calculator
is in degree or radians is when you are touching one of these three buttons, otherwise it does
not make any difference. But if you are using sine, cosine, or tangent, make sure you are
in the proper mode. And for those previous questions, again if they were not one of the
sixteen angles on the unit circle, you would have to radians and now we would have to be
in degree mode. Anyway. The tangent of 270 degrees is y over x. It is negative one over
zero... hmmm We can't divide by zero and that is going to happen, dividing by zero is a
possibility with your four quadrantal angles 0, 90, 180, 270, and 360. You could also go
in the opposite. Any of those quadrantal angles (multiples of 90). So tangent, cosecant, secant,
and cotangent, all four of these trig functions have the possibility of occasionally being
undefined. So make sure you pay attention to that and you know that you cannot divide
by zero. When we go to graph these trig functions, these are going to be where you get your vertical
asymptotes but we are not there yet. Sine and cosine will never become undefined because
they have divisors or denominators of r. A circle cannot have a radius of zero otherwise
it would just be a point and we would not be talking about these trig functions. All
right. BAM!!! Let's do a few in reverse. Let's do a few inverse trig functions. But first
let's pick uh...uh... Actually let's pick an example first. Let's say we want to look
at the cosine of 135. Ok, well the cosine of 135 is the x value of negative square root
of three over two. Excuse me, that is negative square root of two over two. What if I said
what is the inverse cosine of.... Let's put it this way. The cosine of theta is negative
square root of two over two. Now we can see in the previous example the complete equation,
the cosine of 135 is the negative square root of two over two. But what if I left the theta
empty. What if wanted to find what angle measure would give me a cosine value of negative square
root of two over two? Well I would apply the inverse cosine function to both sides of the
equation and theta would be the inverse, with the little negative one, the inverse cosine
of negative square root of two over two. Now in a regular trig function you put in an angle
measurement and you get out the sides of a triangle. With an inverse trig function, and
we will do a video about this later really hitting it hard...right now I am just giving
you a preview, The inverse cosine of negative square root of two over two. See I have the
sides of the triangle and I want the angle. So I need an inverse trig function for that.
Looking on the unit circle, what angle gives me a cosine value of negative square root
of two over two. Well there is an x value of negative square root of two over two, so
we are getting the 135 degrees that I started the question off originally with. But, there
is another point that has a cosine value of negative square root of two over two and that
is 225 degrees or 5pi over 4. So, that is all and dandy but let me just give you another
preview of what is coming up. If you were to put that into your calculator and it was
in degree mode, the calculator would only be able to give you 135 degrees. That is because
you calculator and computers, especially calculators, I am sure there is some kind of computer program
that can handle this...but um.. Calculators can only deal with functions. That means that
when you plug in an x, it can only give you one y. So, be aware that when we start to
get to use our calculators to do inverse trig functions we may not necessarily get the answer
that we want...or all of the answers. And there is more answers to this question. I
just gave you the answers that are between 0 and 360 degrees. You know we can spin around
this unit circle and start talking about coterminal angles and say well there is an indefinite
number of times that I am going to have an x value that is negative square root of two
over two. So we have to restrict where we want the answers from so that you don't just
keep writing for the rest of your life trying to fill up all the answers of the inverse
cosine of negative square root of two over two. Ok... BAM!! Alright. Let's take a look
at one of these reciprocal functions. These functions are not going to be on your calculator
by the way. So if you are asked for a secant, cosecant, or cotangent you will have to pay
special care in how you use your calculator. Or, if it is one of your sixteen angles of
course do it with the unit circle. That is what we are going to do right now. The secant
of negative 7pi over 6. So we have a negative angle measure so that means clockwise rotation
and it is in radians. It has a divisor of 6 so that is multiple of 30 degrees. So, 1,
2, 3, 4, 5, 6, 7pi over 6. If you take negative 7pi over 6 and you add 2pi you will get 5pi
over 6. These are coterminal angles. Secant is.. bababababa... r over x, so it is the
reciprocal of the x coordinate. That means that it is negative two over the square root
of three. You know that we are never going to be happy with radicals in the denominator.
So we are going to multiply the top and bottom by the square root of three. We get negative
two square root of three over three. That is an example of how to one of the reciprocal
functions on the unit circle. I am Mr. Tarrou. That is the unit circle. BAM!!! Go Do Your
Homework.