Professor Dave here, let’s graph some trig functions.
We’ve just spent some time learning about the six trigonometric functions, as well as
how they relate to one another, and the unit circle.
Just like any other function, we will want to be able to graph them, but there are some
interesting things that we can point out about these functions first.
First, let’s talk about the domain and range of the functions.
Looking back at the unit circle, we know that each point has the coordinates cosine theta,
sine theta.
So looking first at sine theta, the domain of this function is the set of all the angles
we could plug into this function, and the range of the function is all the possible
values for sine theta.
This means that the domain of sine theta is all real numbers, since we could plug in any angle.
It could be greater than two pi radians, we just end up going around and around indefinitely
as we approach infinity.
And we could also have negative angles, we just go the other way.
But the range of the function is negative one to one, inclusive.
Sine theta can only have values within this interval, as we can clearly see from the unit
circle, as the Y values of all of these points are somewhere in between negative one and one.
The same goes for cosine theta, we could plug in any angle, but if we look at the X values
of these points, they always fall between negative one and one.
So both sine and cosine will have a domain of all real numbers and a range of negative
one to one.
Tangent is different, because that’s sin over cosine, and as cosine gets very close
to zero, in either direction, the function gets very big, approaching infinity.
So the range of tangent theta will be all real numbers.
The domain is almost all real numbers, but we can’t evaluate tangent when cosine is
zero, because anything over zero is undefined.
Cosine is zero at half pi and three halves pi, so the domain of tangent theta is all
real numbers except half pi plus or minus multiples of pi.
The other thing we want to understand about these trig functions is that they are periodic.
Their values repeat, over and over again, as we go through each period of the function.
The period of these functions is two pi radians, because after two pi radians, we are back
to where we started and all the values repeat.
In other words, the sin of X plus two pi is the same as the sine of X.
The same goes for cosine.
Let’s go ahead and graph the sine and cosine functions now, so we can see exactly what
they look like.
Let’s bring up the coordinate plane, and also the unit circle.
Looking at the Y coordinates of these points, we can start to plot the graph of Y equals
sine X, going in multiples of pi over six.
When X is zero, Y is zero.
When X is pi over six, Y is one half.
When X is pi over three, Y is root three over two.
When X is half pi, Y is one.
Then as we move through quadrant two of the unit circle, the sine values start to go back
down, until we get to X equals pi, where Y is again zero.
Now as we enter quadrant three, we start to get negative values for the Y coordinate,
and thus negative values for sine X.
These will be the same values as the first two quadrants, just negative, until we get
to negative one.
And then moving through the fourth quadrant, we get back to zero.
So there is the graph of Y equals sine X for one period of the function.
Once we get to two pi radians, it’s the same as being at zero, and when we enter another
period of the function, all of the values will repeat.
That’s what makes this a periodic function.
If we graph multiple periods of the function, it looks like this, and we can clearly see
its cyclical nature.
How can we manipulate this function?
Well, we can apply any of the transformations we learned in algebra.
If we put a coefficient here, that will stretch or shrink the function, which will change
the amplitude of the function.
Y equals two sine X will look the same, except that all the values are doubled, so we are
cycling between two and negative two.
If the coefficient is negative, it reflects everything across the X axis, which looks
like this.
In this way, for any function in the form Y equals A sine X, the amplitude will be equal
to the absolute value of A. If instead, there is a coefficient operating on X, that would
be a horizontal stretch, as Y equals sine of two X would mean that the function will
rise and fall at twice the normal rate.
This would mean that the period of this function is pi instead of two pi, and in fact, for
any function in the form of Y equals A sine BX, the period is two pi over B. Lastly, we
can have vertical or horizontal shifts.
If we have a term that is being added to the function, that’s a vertical shift, just
like we saw with parabolas.
Y equals sin X plus one will just shift everything up one.
If instead we have some number inside the function, like Y equals sine of the quantity
X plus half pi, this whole thing will shift half pi to the left.
As it happens, we have just generated the graph for Y equals cosine X.
This will make sense if we refer to the unit circle again, and see that the cosine of zero
is one.
That means the function must start up here.
Then the cosine decreases until we get to zero at half pi, then goes towards negtive
one at pi.
Then it’s back to zero at three-halves pi, and then up to one at two pi, after which
things repeat.
So the graphs for sine and cosine are extremely similar, they are just slightly shifted, and
all of the transformations we used for sine X will apply for cosine as well.
Let’s use what we’ve just learned to graph one period of four sine of the quantity two
X minus two thirds pi.
First let’s find the amplitude.
We can get that from this number here, which is four, so the function will move between
negative four and four.
Then, let’s find the period.
That will be two pi over this term, so the period will be equal to pi.
Then we find the phase shift.
This term would mean that the whole thing is shifted two thirds pi to the right from
the origin, but we have to divide that by this number, since this is causing things
to contract, so that leaves us with pi over three.
So the period will start at one third pi, and end at four thirds pi, going up to four,
and then down to negative four, before coming back to zero.
We can find the X coordinates of these key points as well, we just have to split the
period up into four parts.
That gives us a quarter pi, and we just add quarter pi successively, starting with a third
pi, to get all the X coordinates.
To do that, we need a common denominator, which will be twelve, and then we combine
and reduce, using the rules we already learned for adding fractions.
Before we move on from this subject, let’s just quickly look at the graphs of the other
trig functions so that we know what they look like.
Tangent looks quite a bit different, because the range is no longer negative one to one,
like for sine and cosine.
Instead, this function approaches positive and negative infinity as cosine values approach
zero, and when cosine is zero, tangent is undefined, so we have a series of vertical
asymptotes.
Here is the graph for Y equals tangent X.
It has a period of pi, because tangent values from quadrant one, where sine and cosine are
both positive, come back again in quadrant three, where sine and cosine are both negative.
Likewise, tangent values in quadrant two, which are positive over negative, come back
in quadrant four, where it’s negative over positive.
This function can be transformed in all the same ways that we saw for sine and cosine.
Cotangent will be similar, but as it is the reciprocal of tangent, it’s a little different,
falling to the right instead of rising, and shifted slightly, since the X values that
originally generated asymptotes will now generate Y values of zero, as one over infinity is
zero, and the points that originally had Y values of zero, will now contain asymptotes,
as one over zero is undefined.
Cosecant and secant will also look a little funny, as all the points where the sine and
cosine equal zero will now be asymptotes for their reciprocal functions, again because
one over zero is undefined.
The transformations for all these graphs are exactly what you would expect, so having gone
over this already, let’s check comprehension.