The Amazing Unit Circle
Phase Shift Identities |

The identities on this page are about angles that differ by a right angle. |

Start with the angle θ in standard position. Add a right angle to θ, so angle AOC measures θ + π/2 = θ + 90° and C has coordinates (cos(θ + π/2),sin(θ + π/2)) = (cos(θ + 90°),sin(θ + 90°)). Since angle AOB is a right angle, the angle BOC is &theta. Rotating the angle AOP by a right angle counterclockwise around the origin shows that the length OE = OD = cos(θ) and the length EC = DP = sin(θ). Thus C also has coordinates (-sin(θ),cos(&theta)). Therefore:
or
Two more identities can now be obtained easily. The angle AOF is obtained by
or
Although the diagram shows the angle θ in the first quadrant, the same conclusions can be reached when θ lies in any quadrant, and so these identities hold for all angles θ. The reason for calling these identities Restore initial diagram |

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