The Amazing Unit Circle
Opposite Angle Identities
By opposite angles we mean two angles that differ by a straight angle, that is, by π = 180°.
The angle opposite θ has measure θ + π = θ + 180°. Thus Q has coordinates
The point on the opposite side of the unit circle from (a,b) is the point (-a,-b). So Q also has coordinates (-cos(θ),-sin(θ)). Therefore:
cos(θ+π) = -cos θ &
cos(θ+180°) = -cos θ &
These are the opposite 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 opposite angle identities hold for all angles θ.
Restore initial diagram
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