The Amazing Unit Circle
Complementary Angle Identities |

Two angles are |

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:
or
These are the 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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