# Appendix: Precalculus

**Formulas:**
\[  
e^{ix} = \cos(x) + i\sin(x) && \text{Euler's Identity}
.\]

:::{.proposition title="Angle Sum Identities"}
On one hand,
\[  
e^{i(A+B)} = \cos(A+B) + i\sin(A+B)
,\]
and on the other,
\[  
e^{i(A+B)} 
&= e^{iA} e^{iB} \\
&= \qty{ \cos(A) + i\sin(A) } \qty{\cos(B) + i\sin(B) } \\
&= \qty{ \cos(A) \cos(B) - \sin(A) \sin(B) } + i\qty{\sin(A)\cos(B) + \cos(A) \sin(B) }
.\]

Thus
\[  
\cos(A+B) &= \cos(A) \cos(B) - \sin(A) \sin(B) 
\sin(A+B) &= \sin(A)\cos(B) + \cos(A) \sin(B)
.\]
:::