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Trigonometry Trying to resolve a trig identity

dwsmith

Well-known member
Feb 1, 2012
1,673
I am trying to resolve a trig identity for some notes I am typing up. On paper, I wrote recall $e(\sin(E_1) - \sin(E_0)) = 2\cos(\zeta)\sin(E_m)$. I have no idea what I am recalling this from now at least.

Identities I have set up are:

\begin{align}
E_p &= \frac{1}{2}(E_1 + E_2)\\
E_m &= \frac{1}{2}(E_1 - E_2)\\
x &= a\cos(E)\\
y &= a\sqrt{1 - e^2}\sin(E)\\
\cos(\zeta) &= e\cos(E_p)\\
\alpha &= \zeta + E_m\\
\beta &= \zeta - E_m
\end{align}

Lambert Section
this may be easier to understand if you look at it.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Let's begin with the left side (but write it instead as):

\(\displaystyle e\left(\sin\left(E_1 \right)-\sin\left(E_2 \right) \right)\)

Using a sum-to-product identity, we may write this as:

\(\displaystyle 2e\sin\left(\frac{E_1-E_2}{2} \right)\cos\left(\frac{E_1+E_2}{2} \right)\)

Now using the identities you have set up, this becomes:

\(\displaystyle 2\cos(\zeta)\sin\left(E_m \right)\)