Cosine and Sine Identities with Dihedral Transformations

The relation between certain cosine and sine identities and certain rotations and reflections in the plane are exhibited. For example, is the same as , which is the (90°) clockwise rotation, , of the point .

The cosine and sine angle addition identities encompass the rotation and reflection symmetries of the unit circle. Frequently noted special cases of these identities encompass what is called the dihedral group of index 4. This Demonstration exhibits the relation between these special identities and dihedral symmetries of the unit circle. A (one-dimensional) translation of by , or corresponds to the rotation in the plane by the angle , acting on the point . A (one-dimensional) reflection of across , , , or (given by ) corresponds to the reflection in the plane across the line through the origin at angle (with respect to the positive axis), acting on the point .