The Amazing Unit Circle
Complementary Angle Identities

Two angles are complementary if they add to make a right angle (π/2 = 90°). The angle complementary to θ is the angle π/2 - θ = 90° - &theta.

signs by quadrant

To put the angle complementary to θ in standard position, start by reflecting the angle θ in the line y = x, which bisects the first quadrant. The angle BOQ is θ, so the angle AOQ measures π/2 - θ = 90° - θ. Thus Q has coordinates (cos(π/2-θ),sin((π/2-θ)) = (cos(90°-θ),sin((90°-θ)).

When a point is reflected in the line y = x, its coordinates are reversed. So Q also has coordinates (sin(θ),cos(θ)). Therefore:

cos(π/2-θ) = sin θ & sin(π/2-θ) = cos θ.


cos(90°-θ) = sin θ & sin(90°-θ) = cos θ.

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

Restore initial diagram

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