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4.2 - Video 3 - Applications of the Unit Circle - Duration: 4:03.

In the previous video we obtained this unit circle complete with coordinates for special

angles.

The first application of the unit circle is simply to compute the trigonometric functions

for these angles.

All we need to do is combine the definitions with the diagram and simply match up the coordinates

with the formulas.

For example, if we wanted to compute cotangent of 4 pi/3, we first need to find the angle

on the unit circle and notice that the corresponding x-coordinate is -1/2 and the

corresponding y coordinate is negative -sqrt(3)/2.

Then we look at the formula and see that the cotangent is computed as x over y, so we simply

plug it in and simplify to complete the calculation.

Evaluating any of the six trigonometric functions at any of these angles will follow the same

pattern: identify the angle; determine the coordinates; and then plug it into the

formula.

Since the sine and cosine functions are defined by points on the circle we can see that if

we keep traveling around the circle we're going to repeat the values.

In fact, we know that every time the angle changes by 2 pi we will be back at the exact

same point on the circle.

When something repeats on regular intervals like this we say that it is periodic.

The period is the size of the gap before it repeats.

In this case, the period is 2 pi.

Because we can either add or subtract 2 pi repeatedly, we get a whole collection of values

that correspond to the exact same point on the circle, and since the point gives us the

values of the sine and cosine functions we would say that the sine and cosine functions

are periodic with period 2 pi.

Formally we can express this relationship in the following manner.

The 2 pi n combination corresponds to the increase or decrease in the angle after winding

around an extra n times.

And since the sine and cosine functions determine the rest of the functions, we see that they

all have this property.

We will see these again later on in this course, and for some of these we can actually find

a shorter period.

But we're not going to worry about that for now.

When evaluating the trigonometric functions this relationship allows us to change the

angle by multiples of 2 pi without changing the value of the function itself.

This lets us adjust the angle so that it's back between 0 and 2 pi so that we can use

the angles on the unit circle diagram that we had earlier.

For example, to compute a cosecant of -7 pi/2 we can just keep adding multiples of 2 pi

until we get to a familiar value.

A third application of the unit circle is to understand symmetries into trigonometric

functions.

If we draw the angles plus or minus theta on the unit circle we can see that the x coordinates

of both points perfectly line up.

This is because the upper half and the lower half of the circle are just mirror images

of each other.

This means that if the coordinate of the upper point is (x,y), then the coordinate for the

lower point is (x,-y).

We can plug this into the definitions to discover the following relationships.

Notice that the cosine and secant functions are the only ones that stay the same when

we use the negative angle.

There are two vocabulary terms that are useful here.

We say that a function f is odd if it satisfies the property f(-x) = -f(x) and we say that

a function f is even if it satisfies the property f(-x) = f(x).

In other words the sine, cosecant, tangent, and cotangent functions are odd and the cosine

and secant functions are even.

But it's important not to get hung up into formal mathematical definitions.

The key idea is to think about the geometry of the unit circle.

If you take the point on the unit circle that you get with the angle theta, then the point

on the circle corresponding to negative theta is the reflection of the original point across

the x-axis.

This keeps the x value the same but changes the sign on the y value.

This picture is more important than memorizing that the cosine function is even and the sine

function is odd.