The Amazing Unit Circle
Negative Angle Identities (Symmetry) 
The negative θ of an angle θ is the angle with the same magnitude but measured in the opposite direction from the positive xaxis. A positive angle θ is measured counterclockwise from the positive xaxis, so then θ is measured clockwise from the positive xaxis. 
The negative angle θ is also the angle found by reflecting the angle θ in the xaxis. If the angle AOP is θ, then the angle AOQ is θ. Thus Q has coordinates (cos(θ),sin(θ)). When a point (a,b) is reflected in the xaxis, it moves to the point (a,b). So Q also has coordinates (cos(θ),sin(θ)). Therefore: cos(θ) = cos θ & sin(θ) =  sin θ. These are the negative angle identities. Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the negative angle identities hold for all angles θ. The negative angle identities also tell us the symmetries of the cosine and sine functions. A function f is even if f(x) = f(x) for every x in the domain of f. Since cos(θ) = cos θ, we conclude that cosine is an even function. An even function y = f(x) is symmetric about the yaxis, since for each point (x,f(x)) on the graph of f, the point (x,f(x)) = (x,f(x)) also lies on the graph, so for each x the ycoordinate at x is the same as the ycoordinate at x. A function f is odd if f(x) = f(x) for every x in the domain of f. Since sin(θ) =  sin θ, we conclude that sine is an odd function. An odd function y = f(x) is symmetric about the origin, since for each point (x,f(x)) on the graph of f, the point (x,f(x)) = (x,f(x)) also lies on the graph, so for each x the ycoordinate on the graph at x is the negative of the ycoordinate at x. Restore initial diagram 
The Amazing Unit Circle 
Trigonometry Facts 
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