cosine and sine constructions FINAL
- Author:
- Alec Contreras
The construction of cosine
Begin with what you already had for sine
Make a Segment from Origin O to point Q
Make a line 'a' on Q parallel to the y-axis
Make a point A on the line 'a' and make a segment from the origin to point A, then make a segment from point A to Q to make a triangle.
Select to reflect about the origin.
Construct a line e that is parallel to the x-axis that goes through A'
And then find the intersection between line m and line e and call it C
Why the construction of sin(t) and cos(t) works:
1. The reason these contructions where able to yield us these points is because of the properties of trigonometric identities on the unit circle.
Our coordinates:
A=domain from [1,-1] A'=range[1,-1] Q=[0,2pi] B=(0,pi/2) z=represents the path of sine C=represents the path of cosine
-The unit circle, because it has a radius of one and because it starts off at the origin of the coordinate plane, it can be used to derive trigonometric identies with the construction of a right triangle within the circle.
-The right triangle allows us to manipulate an angle spanning from the origin and then it can be referenced to figure out sine and cosine using the definition of sine on a right triangle=opposite side/hypotenuse and cosine=Adjecent side/hypotenuse.
-In terms of the coordinates because the unit circle or the path of Q represents the movement in radians from 0 to 2pi, hence at q=0 and with sin(q), sin(0)=0 and as Q continues along circle, the output, represented by z gives us all the sine values that are known from [0,2pi].
The same happens with cosine, when point C begins at its origin it yields 1 when Q begins its animation along the unit circle. in other words cos(q) when q=0 is cos(0)=1, and as the radians continue to be inputed cosine continues to yield the path of maximum-0-minimum-0-maximum.