Identities due to Symmetry of the Unit Circle on the origin, x and y axes
Four angles (θ, π  θ, π + θ and 2π  θ) are shown below in a unit circle. To each angle corresponds a point (A, B, C or D) on the unit circle.
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The four angles have the same reference angle equal to θ. Because of the symmetry of the circle, the four points form a rectangle ABCD as shown above. Points A and B are reflection of each other of the yaxis. Points A and C are reflection of each other on the origin of the system of axis. Points A and D are reflection of each other on the xaxis. Given the coordinates a and b of point A and using the symmetries of the circle, the coordinates of A, B, C and D are given by:
A: (a , b) , B: ( a , b), C: ( a ,  b) and D: (a ,  b)
We now express the coordinates of each point in terms of the sine and cosine of the corresponding angle as follows.
A: (a , b) = (cos θ , sin θ)
B: ( a , b) = (cos(π  θ) , sin(π  θ))
C: ( a ,  b) = (cos(π + θ) , sin(π + θ))
D: (a ,  b) = (cos(2π  θ) , sin(2π  θ))
Examples of Identities
Comparing the x and ycoordinates of points A and B, we can write
cos(π  θ) =  cos θ
sin(π  θ) = sin θ
Comparing the x and ycoordinates of points A and C, we can write
cos(π + θ) =  cos θ
sin(π + θ) =  sin θ
Comparing the x and ycoordinates of points A and D, we can write
cos(2π  θ) = cos θ
sin(2π  θ) =  sin θ
More Identities due to Symmetry of the Unit Circle on the x axis (Negative angles)
Two angles θ, and  θ are shown below in a unit circle to which correspond the points A and D on the unit circle.
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Points A and D are reflection of each other on the xaxis. Given the coordinates a and b of point A, the coordinates of D are given by:
D: (a ,  b)
We now express the coordinates of points A and D in terms of the sine and cosine of the corresponding angle as follows.
A: (a , b) = (cos θ , sin θ)
D: (a ,  b) = (cos( θ) , sin( θ))
Examples of Identities that may be Deduced
cos( θ) = cos θ
sin(  θ) =  sin θ
Identities due to Symmetry of the Unit Circle on the line y = x
Points A and B shown in the unit circle below are reflection of each other on the line y = x. Because of the symmetry of the unit circle with respect to the line y = x, the corresponding angles to these points are θ and π/2  θ as shown below.
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Points A and B are reflection of each other on the line y = x. Given the coordinates a and b of point A, the coordinates of B are given by:
B: (b , a)
We now express the coordinates of points A and B in terms of the sine and cosine of the corresponding angle as follows.
A: (a , b) = (cos θ , sin θ)
B: (b , a) = (cos(π/2  θ) , sin(π/2  θ))
Examples of Identities that may be Deduced
cos(π/2  θ) = sin θ
sin(π/2  θ) = cos θ
Use the following general identities
1) cos (A + B) = cos A cos B  sin A sin B
2) cos (A  B) = cos A cos B + sin A sin B
3) sin(A + B) = sin A cos B + cos A sin B
4) sin(A  B) = sin A cos B  cos A sin B
to verify the identities found above and listed below.
Detailed Solutions and explanations to these questions.
 cos(π  θ) =  cos θ
 sin(π  θ) = sin θ
 cos(π + θ) =  cos θ
 sin(π + θ) =  sin θ
 cos(2π  θ) = cos θ
 sin(2π  θ) =  sin θ
 cos( θ) = cos θ
 sin(  θ) =  sin θ
 cos(π/2  θ) = sin θ
 sin(π/2  θ) = cos θ
Detailed Solutions and explanations to these questions.

