Transformation Identities

Identities derived from a $90^o$ shift

Derivation

If we have 2 graphs. One cosine graph, and another sine graph. If the cosine graph was shifted $\frac{\pi }{2}$ degrees to the right, then the 2 graphs would be the same. then it would be an identity $\cos (x-\frac{\pi }{2})=\sin (x)$

Additionally, if the sine graph was shifted $\frac{\pi }{2}$ degrees to the left, the 2 graphs would also be the same. $\sin (x+\frac{\pi }{2})=\cos (x)$

Identities

$\cos (\frac{\pi }{2}-x)=\sin (x)$

$\sin (\frac{\pi }{2}+x)=\cos (x)$