How the Lagna is calculated
The Lagna is calculated from the local sidereal time, the observer's latitude and the obliquity of the ecliptic.
What it is
The Ramc (right ascension of the midheaven) is local sidereal time in degrees. The ascendant tropical longitude is the angle whose cosine and sine follow from the Ramc, the obliquity and the latitude. The standard formula is atan2 of cos(RAMC) and minus (sin(RAMC) cos(e) + tan(latitude) sin(e)).
How it is worked out or read
Subtract the ayanamsa from the tropical ascendant to get the sidereal Lagna. Sidereal time comes from the astronomy engine's Greenwich apparent sidereal time plus the longitude. The site uses the mean obliquity of Meeus.
Where it appears on this site
This is the calculation behind the Lagna in the birth chart. The steps are summarised on the method page.
Step by step for one birth
Take a birth in Mumbai at 06:00 local time on 1 January 1990, which is 00:30 UTC. First the tool finds the Greenwich apparent sidereal time for that moment, which is a number of hours. Multiply by fifteen to convert hours to degrees, then add the longitude of Mumbai, about 72.9 degrees east. The result is the RAMC, the local sidereal time in degrees. Next the obliquity of the ecliptic is computed for the date, a little under 23.44 degrees. With the RAMC, the obliquity and the latitude of 19.08 degrees north, the standard formula gives the tropical ascendant. Finally the ayanamsa for that date is subtracted to give the sidereal Lagna. Each step can be written down and checked, which is what makes the method auditable.
What the formula does not include
The standard formula treats the Earth as a sphere, uses the mean obliquity rather than the true one, and ignores the small effects of refraction on the rising point. For a birth chart these simplifications change the Lagna by at most a few arcminutes, which rarely moves a planet across a sign boundary. A reader who needs more precision should ask a practitioner with a specialised program. The site's Lagna is accurate enough for the house-based rules it supports, and the method page lists each simplification so that nothing is hidden.
Checking your own result
Once you have a Lagna from the tool, you can check it in three ways. First, compare the sidereal time with the value printed in a sidereal almanac for the same date and place. Second, check that the Lagna sign changes at about the right times: if you compute the Lagna for several times a few hours apart, the sign should advance by about one sign every two hours. Third, compare with a second calculator that states its ayanamsa and formula. Agreement within a degree, after matching the ayanamsa, means the result is consistent. Disagreement of several degrees usually points to a difference in the time zone or the coordinates.
Why the method is called a formula
The Lagna calculation is a formula because each step is a fixed arithmetic operation on known quantities: the sidereal time, the latitude, the obliquity and the ayanamsa. Because it is a formula, the result can be repeated exactly by anyone with the same inputs, which is what makes the Lagna checkable rather than a matter of opinion.