Download Radian Measure, Sine and Cosine as points on the Unit Circle, Law

Trigonometry: Radian Measure, Sine and Cosine as points on the Unit Circle, Law of
Cosines, and Trigonometric Identities by Anthony Boyd and Joel Thorne:
length equal to the length of the radius of the circle.
ex:
(1)
ex:
60
(2)
A unit circle is a circle with radius 1 with it's center at the origin. Sine and Cosine are
defined in terms of the coordinates of points that are on the unit circle,
x^2+y^2=1. The Sine function is defined as the vertical coordinate (y) and the Cosine
function is defined as the horizontal coordinate (x).
(black triangle) due to the value of cosine will remain the same, but the value for sine will
become negative, and therefore rotate about the x-axis.
II
III
I
1
IV
ex:
1
2
(3)
1
2
(4)
ex:
(5)
(6)
We also did a proof and example for the Law of Cosines, which states: for length c:
Our example to demonstrate the Law of Cosines also involved use of the Law of Sines,
which states:
Examples of the Trigonometric Identity Formulas in our oral presentation. These
identities include:
A. Addition and Subtraction:
1. sin(x+y)=sinxcosy+cosxsiny
ex:
(7)
at 5 digits
0.96593
(8)
ex:
(9)
at 5 digits
0.96593
(10)
2. sin(x-y)=sinxcosy-cosxsiny
ex:
(11)
at 5 digits
0.25882
ex:
(12)
at 5 digits
0.25883
3. cos(x+y)=cosxcosy-sinxsiny
ex:
(13)
at 5 digits
The formula cos(Pi/3 + Pi/4) changes to -cos(5/12Pi) instead of cos(7/12Pi) is because at
that location on the graph of cos(x) the function rotates about the x-axis. Therefore the
change in the cos(theta) to -cos(theta) shows that the value has went from greater than
1/2Pi to being less than 1/2Pi , and by doing this you must also change the value inside
the cos(theta) to compensate for the change in the negative sign. This changes the value
of cos(7/12Pi) to -cos(5/12Pi).
ex:
(14)
at 5 digits
4. cos(x-y)=cosxcosy+sinxsiny
ex:
(15)
at 5 digits
0.96593
ex:
(16)
at 5 digits
0.96593
B. Double Angle Formula:
1. cos(2x)=cos^2(x)-sin^2(x)
ex:
(17)
ex:
(18)
2. sin(2x)=2sinxcosx
ex:
(19)
ex:
(20)
C. Alternate to the Double Angle:
1. cos(2x)=2cos^2(x)-1
ex:
(21)
ex:
(22)
2. cos(2x)=1-2sin^2(x)
ex:
(23)
D. Half Angle Formula:
1. (1-cos(2x))/2=sin^2(x)
ex:
3
4
(24)
3
4
(25)
1
4
(26)
1
4
(27)
1
2
(28)
1
2
(29)
ex:
2.(1+cos(2x))/2=cos^2(x)
ex:
ex:
E. Conjunction Identities:
ex:
ex:
2. cos(
ex:
(30)
ex:
(31)