Rosh Hashanah is the Rosh Chodesh ("new moon" or "month") of Tishrei.

[MyMission]

1. Another interesting fact about Rosh Chodesh Tishrei

We announce each new month on the Shabbat preceding it, except for Tishrei.
We do not announce it at all. There are several stories about why...some relate to the lunar calendar, not knowing whether a month would have 28 or 29 days, except Ellul, which precedes Tishrei and always has 29 days. So we know when it is and don't need to announce it while the other months are variable.

Another story talks about fooling haSatan as the new year arrives by not announcing it, so he can't cause any trouble.

I never realized until today that we don't announce Tishrei. At services today I noticed we "skipped" it, so I asked the rabbi. Got those 2 stories.

About the moon, a number of years back, on Passover, I saw the full moon and something clicked. The moon is always full on Passover. And I then realized it was also full on Sukkot and Purim too. The holidays that fall on the 14th or 15th all have a full moon.