Video Transcript
In this video, we’ll learn how to
graph trigonometric functions such as sine, cosine, and tangent and deduce their
properties. We’ll learn how to apply simple
transformations to graph our functions in this form and recognize the relationship
between each graph and the unit circle. Here is a unit circle. It’s a circle with a radius of one
unit, which we’ve plotted with its center at the origin of a pair of 𝑥𝑦-axes. Let’s add a point 𝑃, which can
move around circumference of the circle. We could call its coordinates 𝑥,
𝑦. But what if we then add a
right-angle triangle to our diagram and define an angle here. This is called the included angle,
and we’re going to call it 𝜃.
At the moment, we see that the
length of the base of the triangle must be 𝑥 units, and its height must be 𝑦
units. We could, however, use trigonometry
to find expressions for 𝑥 and 𝑦 in terms of 𝜃. We use the standard convention for
labeling right-angled triangles. The side opposite the included
angle is the opposite side. The longest side, that’s the one
that sits opposite the right angle, is the hypotenuse. And the other side, the side
between the right angle and the included angle, is the adjacent. We recall the acronym
SOHCAHTOA. And we see that we can use the
cosine ratio to find a link between 𝑥 and 𝜃. This is cos of 𝜃 is adjacent over
hypotenuse.
Well, the adjacent side in our
triangle is 𝑥 units and the hypotenuse is one unit, so cos 𝜃 is 𝑥 over one or
simply cos 𝜃 is equal to 𝑥. Similarly, we’ll use the sine ratio
to link 𝑦 and 𝜃. This time, the opposite is 𝑦 and
the hypotenuse is one. So, we get sin 𝜃 is 𝑦 divided by
one, or simply sin 𝜃 is equal to 𝑦. We can now say that the point 𝑃
must have coordinates cos 𝜃, sin 𝜃. Now, as we move 𝑃 around the
circumference of the circle in an anticlockwise direction, the size of 𝑥 and 𝑦,
and therefore the size of cos 𝜃 and sin 𝜃, will change. In fact, they do something really
interesting. Let’s see what that is.
Let’s imagine our point lies on the
positive 𝑥-axis here. At this point, 𝜃 is zero. And the point has coordinates one,
zero. When 𝜃 is zero, then, we see that
cos of 𝜃, which represents 𝑥, is equal to one. sin of 𝜃, which represents 𝑦, is equal to zero. We’ll now repeat this process
here. This time, 𝜃 is equal to 45
degrees. And we can add a right-angled
triangle and we see the hypotenuse is equal to one unit. However, since this is a
right-angled triangle with one angle of 45 degrees, we know in fact it’s an
isosceles triangle. And so, we can say that its other
two sides are equal to 𝑎 units.
The Pythagorean theorem tells us
that the sum of the squares of the two shorter sides is equal to the square of the
longer. That is, 𝑎 squared plus 𝑎 squared
equals one squared or two 𝑎 squared is equal to one. We divide through by two, and we
find 𝑎 squared is equal to one-half. And then we find the square root of
both sides. Now, 𝑎 is a length, so we’re going
to say that it’s the positive square root of one-half, which we can write as root
two over two. And so, we see when 𝜃 is equal to
45 degrees, our point has coordinates root two over two, root two over two. This tells us that sin of 𝜃 when
𝜃 is 45 degrees is root two over two, as is cos 𝜃.
And what about up here? Well, this time 𝜃 is 90
degrees. And of course, our circle has a
radius of one. So, this point has coordinates
zero, one. And we see then when 𝜃 is 90
degrees, cos of 𝜃, which is the 𝑥-value, is zero and sin of 𝜃, which is the
𝑦-value, is one. We can repeat this process for when
𝜃 is equal to 180 degrees. At this point, we have a point with
coordinates negative one, zero. So, when 𝜃 is 180 degrees, cos of
𝜃 is negative one and sin of 𝜃 is zero. Then, when 𝜃 is 270 degrees, our
point has coordinates zero, negative one. So, sin of 𝜃 is negative one,
remember that’s the 𝑦-coordinate, and cos of 𝜃 is zero.
And then, we continue around the
circle and we see we get back to the start. So, when 𝜃 is equal to 360
degrees, we have the same values for sin 𝜃 and cos 𝜃 as we did when 𝜃 was equal
to zero. sin 𝜃 is zero, and cos 𝜃 is one. Let’s fill in these gaps. At this point, 𝜃 is 135
degrees. If we add in a right-angled
triangle and use the fact that angles on a straight line sum to 180 degrees, we see
we have a replica of the earlier triangle we looked at. It has a hypotenuse of one unit and
an included angle of 45 degrees. This means the lengths of its other
two sides must be root two over two units. In coordinate form then, this point
must have coordinates negative root two over two, root two over two. And so, when 𝜃 is 135 degrees, cos
of 𝜃, which is our 𝑥-coordinate, is negative root two over two, and sin 𝜃 is root
two over two.
When 𝜃 is 225 degrees, we have a
similar situation. We have another isosceles triangle
with a hypotenuse of one unit and two other sides of root two over two units. This means when 𝜃 is 225 degrees,
our point must have coordinates negative root two over two, negative root two over
two, meaning sin of 𝜃 and cos of 𝜃 are both negative root two over two. Finally, when 𝜃 is 315 degrees, we
have another isosceles triangle. This time, the coordinates of our
point are root two over two and negative root two over two, meaning that when 𝜃 is
315 degrees, sin 𝜃 is negative root two over two and cos 𝜃 is root two over
two.
Now, it should be quite clear that
if we were to continue moving around this circle in an anticlockwise direction, we
would meet all of these points once again. So, when 𝜃 is 405 degrees, sin of
𝜃 and cos of 𝜃 are the same when 𝜃 is equal to 45 degrees and so on and this
leads us to a definition. We say that sin of 𝜃 and cos of 𝜃
are periodic functions; that is, they repeat. They do so every 360 degrees. So, we say their period is 360
degrees. Let’s sketch the graphs. We’ve seen that both graphs
oscillate; that is, they move between the values of one and negative one. Those are their maximums and
minimums.
So, we’ll start with the graph of
sin of 𝜃 or, in this case, 𝑦 is equal to sin of 𝑥. It passes through the origin; that
is, when 𝜃 is zero, sin of 𝜃 is zero. So, when 𝑥 is zero, sin of 𝑥 is
zero. Then, when 𝑥 is 45, we get sin of
𝑥 to be root two over two. That’s roughly here. When 𝑥 is 90, sin of 𝑥 is
one. Then, when 𝑥 is a 135, sin of 𝑥
is root two over two. And when 𝑥 is 180, sin of 𝑥 is
zero. We could continue in this
manner. Let’s join these up with a smooth
curve. And when we do, we find the graph
of 𝑦 equals sin of 𝑥 between the values of zero and 360 looks a little bit like
this. Of course, we said that these
graphs are periodic. They repeat, so we could continue
in this manner, repeating this exact graph every 360 degrees.
Similarly, the graph of 𝑦 equals
cos 𝑥 in the interval from zero to 360 looks like this. Since it is also periodic, we could
continue either side in the same manner. And so, to recap a few features of
our graphs, they’re periodic; they have a period of 360 degrees. They both have maxima at one and
minima at negative one. Both of them, in fact, have a
little bit of symmetry. The graph of 𝑦 equals cos of 𝑥
has reflectional symmetry about the line 𝑥 equals 180 or the line 𝑥 equals
zero. The graph of 𝑦 equals sin of 𝑥
has rotational symmetry about the origin. But if we narrow it down and just
look at, say, a part of the graph, that is, the interval from zero to 180, we see
that 𝑦 equals sin of 𝑥 does have a little bit of reflectional symmetry about the
line 𝑥 equals 90 degrees.
These features can help us to solve
problems involving cos and sin of 𝑥. The graph of 𝑦 equals tan of 𝑥 is
a little bit stranger than this. And rather than using the unit
circle method, we’re going to plot a table of values using our calculator. For values of 𝜃 of zero, 45, 90,
135, and so on, we get the following table of values. Notice we have an error at 𝜃
equals 90 degrees and 𝜃 equals 270 degrees. This will continue every 180
degrees. But what’s actually happening
here? Well, we know tan 𝜃 is opposite
over adjacent, but we can’t actually draw a right-angled triangle at 𝜃 equals 90
degrees. And so, we say that as 𝜃
approaches 90 degrees, tan of 𝜃 approaches ∞. We can’t evaluate it, and so we
represent this fact using vertical asymptotes on our graph.
The graph of 𝑦 equals tan 𝜃 gets
closer and closer to these asymptotes but will never quite touch them. And so, the graph of 𝑦 equals tan
of 𝑥 looks a little something like this. Once again, we see that the
function tan of 𝜃 is periodic. But this time, its period is 180
degrees. It has rotational symmetry about
the origin. And we can’t define its maximums or
minimums because we saw that as 𝜃 approaches 90 and then multiples of 180 degrees,
tan of 𝜃 approaches ∞.
It’s really important that we can
identify and sketch the graphs of 𝑦 equals sin of 𝑥, 𝑦 equals cos of 𝑥, and 𝑦
equals tan of 𝑥. And we also need to be able to
transform them such that for the graph of 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑥
plus 𝑎 is a translation by the vector negative 𝑎, zero, whereas 𝑦 equals 𝑓 of 𝑥
plus 𝑏 is a translation by the vector zero, 𝑏. 𝑦 equals 𝑓 of 𝑎 times 𝑥 is a
horizontal stretch with a scale factor one over 𝑎. And 𝑦 equals 𝑏 times 𝑓 of 𝑥 is
a vertical stretch by a scale factor of 𝑏. Then, 𝑦 equals negative 𝑓 of 𝑥
is a reflection in the 𝑥-axis, and 𝑦 equals 𝑓 of negative 𝑥 is a reflection in
the 𝑦-axis.
Finally, sometimes we measure
angles in radians such that two 𝜋 radians is equal to 360 degrees and 𝜋 radians is
180 and so on. Now, don’t worry if you’ve not
encountered these yet, they’re just another way of representing an angle. So, let’s have a look at some
questions on trigonometric graphs.
Assign each plot shown in the graph
below to the function it represents.
Here, we see we have two very
similar-looking graphs. These graphs are clearly
periodic. They appear to repeat every two 𝜋
radians. Remember, that’s repeating every
360 degrees. We see they have maximums at one
and minimums at negative one, respectively. In fact, we know that we can
describe the graphs of the sine and cosine functions in the same way.
The main difference is where these
graphs intersect the 𝑦-axis. 𝑦 equals sin of 𝑥 passes through
the 𝑦-axis at zero, whereas 𝑦 equals cos of 𝑥 passes through at one. And we said, of course, that both
have a period of 360 degrees or two 𝜋 radians, maxima at one, and minima at
negative one. This means that the red plot which
intersects the 𝑦-axis at one must be the cosine curve, whereas the blue plot must
be the sine curve.
Now, this actually shows a really
interesting feature of these functions. Let’s define 𝑓 of 𝑥 to be equal
to sin of 𝑥. Then, we see that the function 𝑓
of 𝑥 plus 𝜋 over two or 𝑓 of 𝑥 plus 90, which represents a translation by the
vector negative 90, zero, maps the sine curve onto the cosine curve. And of course, the converse is also
true. So, we’ve seen that there’s a
relationship between the sine and cosine graphs by a horizontal translation. Now, let’s look at a stretch.
Find the maximum value of the
function 𝑓 of 𝜃 is equal to 11 sin 𝜃.
Firstly, we’re going to recall what
the graph of 𝑓 of 𝜃 equals sin of 𝜃 looks like. It has maxima and minima at one and
negative one, respectively. We know that it passes through the
origin and that it’s periodic and it has a period that repeats every 360
degrees. So, 𝑓 of 𝜃 equals sin of 𝜃 has a
graph that looks a little something like this. But of course, we were actually
interested in the graph of the function 𝑓 of 𝜃 is 11 sin 𝜃.
And so, we recall that for a
function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑎 times 𝑓 of 𝑥 represents a vertical
stretch by a scale factor of 𝑎. In this case, we can see our scale
factor is 11. And so, 𝑓 of 𝜃 equals 11 sin 𝜃
looks something like this. It still intersects the 𝑥- and
𝑦-axes at the same places, but now it travels as high as 11 and as low as negative
11. And so, the maximum value of the
function 𝑓 of 𝜃 equals 11 sin 𝜃 is 11.
We’ll now have a look at a
reflection.
Which of the following is the graph
of 𝑦 equals negative tan of 𝑥?
Let’s just begin by recalling what
the graph of 𝑦 equals tan of 𝑥 looks like. It’s periodic, and it repeats every
180 degrees. It passes through the origin, the
point zero, zero. It has vertical asymptote at 𝑥
equals 90 degrees but also every 180 degrees either side of this, in other words, 𝑥
equals negative 90, 𝑥 equals 270, and so on. In fact, the graph of 𝑦 equals tan
of 𝑥 is this one. It’s (A).
Notice that the graph approaches
the asymptotes but never actually quite touches them. But of course, we were interested
in the graph of 𝑦 equals negative tan of 𝑥. So, we recall that for a function
𝑦 equals 𝑓 of 𝑥, 𝑦 equals negative 𝑓 of 𝑥 is a reflection in the 𝑥-axis. And we see that the only graph that
matches this is (D). (D) is the graph of 𝑦 equals
negative tan of 𝑥.
In this video, we’ve learned what
the graphs of 𝑦 equals sin of 𝑥, 𝑦 equals cos of 𝑥, and 𝑦 equals tan of 𝑥
looks like. We saw that the graphs of 𝑦 equals
sin of 𝑥 and 𝑦 equals cos of 𝑥 are periodic; they have a period of 360 degrees,
whereas the graph of 𝑦 equals tan of 𝑥 repeats every 180 degrees. Finally, we saw that 𝑦 equals sin
of 𝑥 and cos of 𝑥 have maxima and minima at one and negative one, respectively,
whereas the graph of 𝑦 equals tan of 𝑥 has asymptotes at 𝑥 equals 90 and then
multiples of 180 degrees.
