The definition of the cos for any angle is the length of the x side of the right triangle for
that angle in a unit circle.
The definition of the sin for any angle is the length of the y side of the right triangle for
that angle in a unit circle.
It's that simple.
This means that the coordinates of any point (x, y) on a unit circle can also represented
as ( cos(θ), sin(θ) ).
In programming, when we want to position an object at any point on a unit circle, we:
(a) Calculate the Angle, then
(b) Use cos and sin as the x and y coordinates.
Suppose, however, that we want to position objects on a circle whose radius does not
equal 1.
If we let r represent the radius, and we know the angle,
then the coordinates of the corresponding point on the circle will
be ( r x cos(θ), r x sin(θ) ).
Why?
In the diagram below, you can see the smaller unit circle within a large circle with radius
= r.
The two circles are CONCENTRIC, meaning they share the same center point.
The two triangles are SIMILAR, meaning that their angles are all the same, even
though they are different sizes.
In similar triangles, sides which correspond to each other arePROPORTIONAL.
That means:
If you multiply one side of the small triangle by a factor
and get the length of the corresponding side in the large triangle,
then you can multiply the other two sides of the small triangleby the same factor
and get the lengths of the remaining two corresponding sides in the large triangle.
So ... since the length of the hypotenuse of the unit circle = 1,
we simply multiply by r to get the length of the hypotenuse of the large circle.
To calculate the length of the horizontal leg of the large triangle (x),
you also multiply the length of the horizontal leg of the small triangle cos(θ) by r,
which gives r x cos(θ).
To calculate the length of the vertical leg of the large triangle (y),
you also multiply the length of the vertical leg of the small triangle sin(θ) by r,
which gives r x sin(θ).
Free Response Questions Assignment (short answer):
In your BYOB/02Astronomy/02RotatingEarth folder, create an MSWORD document called FRQ-Circle.doc (or .docx).
Answer the following questions:
1. The diagram on the left shows a unit circle. What would be the coordinates of the 4
points (A-D) if the radius of the circle were 7.6 ?
2. What would be the coordinates of the 4 points (A-D) if the circle's radius = r?
3. What are the coordinates of point E ?
4. What would be the coordinates of point E if the circle's radius = 3 ?
5. What would be the coordinates of point E if the circle's radius = r ?
ANSWERS:
1.___________________________________________________________________________
2.___________________________________________________________________________
3.___________________________________________________________________________
4.___________________________________________________________________________
5.___________________________________________________________________________