Mathematical Identities
Contents
Mathematical Identities#
Trigonometric identities#
\(\cos(\cdot)\) to \(\sin(\cdot)\)#
\[\begin{align*}
A \cos(\omega t + B) &= A \sin(\frac{\pi}{2} - (\omega t + B))\\
&= -A \sin(\omega t + B -\frac{\pi}{2})
\end{align*}\]
\(\sin(\cdot)\) to \(\cos(\cdot)\)#
\[\begin{align*}
A \sin(\omega t + B) &= -A \cos(\omega t + B +\frac{\pi}{2})
\end{align*}\]
Trigonometric Derivatives#
Derivative of \(\cos(\omega t + \theta)\)#
\[\begin{align*}
\frac{d}{dt} \left ( \cos(\omega t + \theta) \right ) = - \omega \sin(\omega t + \theta)
\end{align*}\]
Derivative of \(\sin(\omega t + \theta)\)#
\[\begin{align*}
\frac{d}{dt} \left ( \sin(\omega t + \theta) \right ) = \omega \cos(\omega t + \theta)
\end{align*}\]
Trigonometric Integrals#
Integral of \(\cos(\omega t + \theta)\)#
\[\begin{align*}
\int \cos(\omega t + \theta) dt &= \frac{1}{\omega} \sin(\omega t + \theta) + c
\end{align*}\]
Integral of \(\sin(\omega t + \theta)\)#
\[\begin{align*}
\int \sin(\omega t + \theta) dt &= - \frac{1}{\omega} \cos(\omega t + \theta) + c
\end{align*}\]
Adding two cosine waves of the same frequency#
\[\begin{align*}
A \cos(\omega t + B) + C \cos(\omega t + D) &= \Re\{A e^{j (\omega t + B)} + C e^{j(\omega t + D)}\}\\
&= \Re\{e^{j\omega t}(A e^{j B} + C e^{jD})\}\\
&= \Re\{e^{j\omega t} ( A \cos(B) + C \cos(D) \\
&+ j (A \sin(B) + C \sin(D))) \}\\
&= \Re\{e^{j\omega t} \\
& \small \sqrt{(A \cos(B) + C \cos(D))^2 + (A \sin(B) + C \sin(D))^2} \\
& e^{j \arctan(A \sin(B) + C \sin(D), A \cos(B) + C \cos(D))}\}
\end{align*}\]
which yields
\[\begin{split}
\sqrt{(A \cos(B) + C \cos(D))^2 + (A \sin(B) + C \sin(D))^2} \\ \cos(\omega t + \arctan(A \sin(B) + C \sin(D), A \cos(B) + C \cos(D)))
\end{split}\]