Two angles are supplementary if they add to make a straight angle (π = 180°). The angle supplementary to θ is the angle π - θ = 180° - &theta.

To put the angle supplementary to θ in standard position, start by reflecting the angle θ in the y-axis. The angle BOQ is θ, so the angle AOQ measures π - θ = 180° - θ. Thus Q has coordinates (cos(π-θ),sin(π-θ)) = (cos(180°-θ),sin(180°-θ)).

When a point (a,b) is reflected in the y-axis, it moves to the point (-a,b). So Q also has coordinates (-cos(θ),sin(θ)). Therefore:

cos(π-θ) = - cos θ & sin(π-θ) = sin θ

or

cos(180°-θ) = - cos θ & sin(180°-θ) = sin θ.

These are the supplementary 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 supplementary angle identities hold for all angles θ.

Restore initial diagram