When we go to an amusement park and board a ferris wheel, we realize a periodic
movement, where the movement is repeated over and over again.
To describe this periodic mathematical movement, we need a function
whose values increase, then decrease and repeat the pattern
indefinitely. To begin exploring this idea, in this lesson, we use movement on the
unit circle and define the functions sine and
cosine. In this section, we will discuss the properties of these
functions and analyse their graphs.
To study movement and locations in circles it is convenient to
initially set ourselves in the simple circle.
The unit circle is the circle with a
radius of 1, centered in the origin of the Cartesian plane.
It's equation is:
There are many ways to define the locations in a unit circle.
We are going to begin defining a location (x,y)
in the circle by the angle formed by (x,y) and (0,0)
and (0,0) and (1,0).
Like we saw in the previous section, there exists a correspondence
with a point (x,y) of the unit circle and the
angle with vertices in the origen and in the point
terminals(x,y) y (1,0). This correspondence permits
us to define the following functions:
The following application allows us to see the associated
location to a point in the unit circle. To see the functions
sin(α) and cosine(α) mark the corresponding boxes.
The following table shows some important properties of the functions
sine and cosine.
For any point (x,y) in the unit circle, we can construct a right triangle like is shows in the graph to the left. Note that the sides corresponds to the coordinates x and y of the point respectively. The hypotenuse is the radius of the unit circle.
The following table shows the functions
sine and cosine of
some angles.
From the figure we can see that the angle with measure 0° is associated with the point (1, 0)
Hence:
and
We want to get the coordinates of the point associated with a 30° angle. This coordinates coincide with the measure of the sides of the triangle shown in the figure.
Since the unit circle has radius equal to 1 the hypotenuse of the triangle must be equal to one.
Reflecting the triangle on one of its sides:
Since the addition of the angles of a triangle equals 180°, the other angles measure 60°, hence we have an equilateral triangle.
Each side opposite to the angle 30° measure ½
Now we know that y equals ½. To find x we will use the Pythagorean Theorem:
This implies that the sides of our triangle has the following measures:
From where we can conclude that the angle of 30° is associated with the point
Hence:
and
To get the coordinates associated with an angle of 45°, we will work with the triangle shown in the figure. Since the radius of the unit circle equals 1 the hypotenuse of the triangle must equal one.
Any right triangle with one of its angles measuring 45° is an isosceles triangle, in other words, the two smaller sides has the same length.
Using the Pythagorean Theorem:
Since a Must be a positive value:
This implies that the sides of our triangle have the following measures:
From where we can conclude that, an angle of 45° is associated with the point
Hence:
and
We want to get the coordinates associated with an angle of 60°. This coordinates coincide with the sides of the right triangle shown in the figure.
Since the radius of the unit circle equals 1 the hypotenuse of the triangle must equal one.
Reflecting the triangle over one of its sides.
Since the addition of the angles of a triangle equals 180°, the other angle must equal 60° and aan equilateral triangle is formed.
Hence, the base of both smaller triangles measure ½.
Now we know that x equals ½. To find y we will use the Pythagorean Theorem:
This implies that the original triangle have the following dimensions:
From where we can conclude that the angle of 60° is associated with the point
Hence:
and
The figure shows an angle of 90° it is associated with the point (0, 1)
Hence:
and
We see that:
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 60° we have that:
Hence:
and
We see that:
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 45° :
Hence:
and
We see that:
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 30° :
Hence:
and
In the figure we can see the angle measuring 180° this one is associated with the point (-1, 0)
Hence:
and
We see that:
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 30° :
Hence:
and
We see that:
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 45° :
Hence:
and
We see that:
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 60° :
Hence:
and
In the figure we can see the angle measuring 270° this one is associated with the point (0, - 1)
Hence:
and
We see that:
Angles measuring 300° and 60° are associated with the same location.
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 60° :
Hence:
and
We see that:
Angles measuring 315° and 45° are associated with the same location.
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 45° :
Hence:
and
We see that:
Angles measuring 330° and 30° are associated with the same location.
On the other hand, in the properties described in the previous section, we saw that:
and
Applying those properties and the results obtained for the sine and cosine of 30° :
Hence:
and
Functions Sine and
Cosine
Click on any of the
Angles
Graph
Demonstration
To practice exercises about sine and cosine of special angles, click
on the following button
Remember that an angle is formed by one initial side, one terminal
side, and the vertice, as shown in the figure:
In the system of cartesian coordinates, it is said that one angle is
in the standard position when the vertice is in the origin and the
initial side is in the positive side of the line x.
Examples:
As we saw in the above examples, the positive angles are measured
counterclockwise; the negative angles are measured in the clockwise
sinse.
Coterminal Angles
The angles that have the same sides, are called coterminal
angles.
Examples:
Quadrant Angles
The angles in the standard position through the terminal side are in
one of the coordinating axis are called quadrants.
Examples:
Angles of Reference
The angle of reference α' of an angle α
in the standard position, is the acute angle formed by the sides of
α and the axis x.
Examples:
The values of the trigonometric functions of angles more than
90° ( or less than 0°) can be determined from their
angles of reference. In this case, one should be able to have
the corresponding sign to the quadrant where the angle is
located.
Remember hat we can define angles that are described more than a
return around the circle. The values of the functions sine and cosine
are the same for coterminal angles, for example sin(390°) =
sin(30°). In this way the values of the function are repeated
periodically in an indefinite form.
A function is periodical if its
graph is repeated every interval of the independent
variable, period.
Examples:
1.
The following function is periodical, but it sgraph is
repeated every 180 units. It's to say, it has a period of
180.
2.
The sin(x) function is periodocial, but their graph
repeats every 360°. In this case we say that it's term is
360°
3.
The function cos(x) is periodical, but it's graph is
repeated every 360°. In this case we say that it's period
is of 360°
It's important to nte that in every period the graph is exactly the
same. The following function, for example, is not periodical:
In the following application we can appreicate the graph of the
function cosine. Observe that the periodical of the cosine function is
360°
Now that you have completed this less, you should be able to:
Identify a unit circle and its relation with the real
numbers.
Evaluate trigonometric functions using the unit circle.