<p>That's trigonometry:</p>
<p><span class="math-container">$$\sin (\alpha+\beta) = \sin\alpha\cos\beta + \sin\beta\cos\alpha$$</span>
or simplified with <span class="math-container">$\alpha = \beta$</span> to
<span class="math-container">$$\sin 2\alpha = 2\sin\alpha\cos\alpha$$</span></p>
<p>This identity is applied to the taylor expansion mentioned above in the text, giving the results quoted.</p>
<p>Check out, for instance, <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities" rel="noreferrer">https://en.wikipedia.org/wiki/List_of_trigonometric_identities</a> for a list of useful trigonometric identities.</p>

CC BY-SA 4.0
2024-09-13T07:38:38.770Z
Arrrrrrrrrrrgh That's so obvious now that you say it! I need to take a rest! Thanks a lot
Vincent ISOZ
CC BY-SA 4.0
2024-09-13T07:17:18.680Z
2024-09-13T11:38:34.693Z
2024-09-13T11:38:34.693Z
planetmaker
<p>In the document available here:</p>
<p><a href="http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.pdf" rel="nofollow noreferrer">http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.pdf</a></p>
<p>I have been able so far to reproduce all missing details so far to the relation (33) of page 7:</p>
<p><span class="math-container">$$\sin E=\frac{E-M}{e}=\sin M+\frac{e}{2} \sin 2 M+\frac{e^2}{8}(3 \sin 3 M-\sin M)+\ldots$$</span></p>
<p>However (it must be very likely obvious but i don't get it...) i don't understand how from (33) the author get (34), ie:</p>
<p><span class="math-container">$$\sin 2 E=\sin 2 M+e \sin M 2 \cos 2 M+\ldots$$</span></p>
<p>Any help/advice is greatly appreciated.</p>

<the-sun><time><analemma>
Equation of Time (derivation Analemma)
Vincent ISOZ
# Equation of Time (derivation Analemma)

(Asked by: username_0 on 2024-09-13)

In the document available here:

<http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.pdf>

I have been able so far to reproduce all missing details so far to the relation (33) of page 7:

$$\sin E=\frac{E-M}{e}=\sin M+\frac{e}{2} \sin 2 M+\frac{e^2}{8}(3 \sin 3 M-\sin M)+\ldots$$

However (it must be very likely obvious but i don't get it...) i don't understand how from (33) the author get (34), ie:

$$\sin 2 E=\sin 2 M+e \sin M 2 \cos 2 M+\ldots$$

Any help/advice is greatly appreciated.

**Tags:** the-sun, time, analemma

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## 1 answers

### Answer by username_1 on 2024-09-13. Score: 8 | Accepted answer

That's trigonometry:

$$\sin (\alpha+\beta) = \sin\alpha\cos\beta + \sin\beta\cos\alpha$$
or simplified with $\alpha = \beta$ to
$$\sin 2\alpha = 2\sin\alpha\cos\alpha$$

This identity is applied to the taylor expansion mentioned above in the text, giving the results quoted.

Check out, for instance, <https://en.wikipedia.org/wiki/List_of_trigonometric_identities> for a list of useful trigonometric identities.

> **Comments:**
> **username_0**: Arrrrrrrrrrrgh That's so obvious now that you say it! I need to take a rest! Thanks a lot  

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