Cosine and Sine Identities with Dihedral Transformations

rotation

0

π/2

π

-π/2

reflection

y = 0

y = x

x = 0

y = -x

angle t

2.92

dihedral transformation: (x, y) ↦ (y, x)

identities: (cos(π/2 - t), sin(π/2 - t)) = (sin(t), cos(t))

The relation between certain cosine and sine identities and certain rotations and reflections in the plane are exhibited. For example,

(cos(t+π/2),sin(t+π/2))

is the same as

(-sin(t),cos(t))

, which is the

π/2

(90°) clockwise rotation,

(x,y)↦(-y,x)

, of the point

(cos(t),sin(t))

.