The negative -θ of an angle θ is the angle with the same magnitude but measured in the opposite direction from the positive x-axis. A positive angle θ is measured counterclockwise from the positive x-axis, so then -θ is measured clockwise from the positive x-axis.

The negative angle -θ is also the angle found by reflecting the angle θ in the x-axis. 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 x-axis, 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 y-axis, 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 y-coordinate at -x is the same as the y-coordinate 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 y-coordinate on the graph at -x is the negative of the y-coordinate at x.

Restore initial diagram