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Trigonometric identities can feel like a list to memorise. For Indian astronomers 1,500 years ago they were tools for building tables they needed every day. Varahamihira's Pancha Siddhantika shows this clearly: it gives a rule for computing sines from other sines and a full table built with it. This post covers who Varahamihira was, what the Pancha Siddhantika contains, and a worked example of his sine rule that you can check against your calculator.
Who was Varahamihira?
The MacTutor biography opens with a warning: "Our knowledge of Varahamihira is very limited indeed." It gives his dates as 505 to 587. He was educated in a place called Kapitthaka, whose location is debated, and he worked at Ujjain, which had been a centre for mathematics since around 400 CE. MacTutor notes that Brahmagupta was the next major figure of the Ujjain school.
MacTutor also calls him one of the most famous astrologers in Indian history. His Brihat Samhita covers the movements of heavenly bodies, weather, omens and much else.
What is the Pancha Siddhantika?
The name means "five siddhantas", or five astronomical treatises. The 1889 edition and translation by George Thibaut and Sudhakara Dvivedi, available on the Internet Archive, lists them as the Paitamaha, Vasishtha, Romaka, Paulisa and Saura (Surya) Siddhantas. Thibaut explains that Varahamihira ranked them: the Surya Siddhanta first, the Romaka and Paulisa about equally correct, and the other two far behind. He gave them space to match, with the Surya and Paulisa treated at length and the Paitamaha very briefly. MacTutor points out that several of these older texts are now lost, so the Pancha Siddhantika is also our source of information about them. Thibaut notes that it refers in one place to the views of Aryabhata, whose own work dates from 499 CE; see our post on Aryabhata's astronomy.
When was the Pancha Siddhantika written?
This is a good example of how historians reason about dates. The book's day-count rule tells you to subtract 427 from the current year of the Saka era, so its calculations start from Saka 427, which is 505 CE. Some 19th-century scholars took that as Varahamihira's birth year. Thibaut disagreed: 505 is the starting point of the calculations, which may have been borrowed from the Romaka Siddhanta, not necessarily the year the book was written. He concluded that it was composed between 505 and 587, "probably" around the middle of the 6th century. MacTutor dates it to 575.
How did Varahamihira compute sines?
Indian astronomers worked with the sine of an arc in a circle of a chosen radius R, so their "sine" was R sin θ in modern terms. Thibaut explains that the fourth chapter of the Pancha Siddhantika uses R = 120 and lists sines at steps of 3°45′, twenty-four values for the quarter circle. Stanza 5 of that chapter gives a short rule: take the radius minus the sine of (90° minus twice the arc), multiply by 60, and you get the square of the sine of the arc.
In modern symbols, with R = 120:
(R sin θ)² = 60 × (R − R sin(90° − 2θ)) = 60 × (120 − 120 cos 2θ).
Divide both sides by 120² = 14,400 and this becomes sin²θ = ½(1 − cos 2θ). That is the half-angle identity from your Class 11 textbook, and MacTutor lists it among Varahamihira's formulas, along with sin²x + cos²x = 1.
Worked example: Varahamihira's sine of 15°
We want R sin 15° with R = 120.
Find 90° − 2θ. 90° − 30° = 60°.
Find R sin 60°. You can get this from Pythagoras, as the text does: R sin 60° = √(R² − (R/2)²) = √(14,400 − 3,600) = √10,800 = 103.923.
Subtract from the radius. 120 − 103.923 = 16.077.
Multiply by 60. 60 × 16.077 = 964.62.
Take the square root. √964.62 = 31.058.
Now check it against the table. In Thibaut's translation, the first eight sines for Aries (up to 30°) are 7;51, 15;40, 23;25, 31;4, 38;34, 45;56, 53;5 and 60;0, in units of minutes and seconds of the radius. The fourth entry, for 15°, is 31;4, that is 31 + 4/60 = 31.067. Our value is 31.058, and a calculator gives 120 sin 15° = 31.058. The table is correct to within about half a sixtieth of a unit.
What mistakes do students make with half-angle formulas?
Using the wrong double angle. The rule needs cos 2θ, not cos θ. For θ = 15° you need cos 30°, which is the same as sin 60°.
Forgetting the square root. The rule gives the square of the sine. Students often stop at 964.62.
Reading sexagesimal as decimal. 31;4 means 31 and 4/60, not 31.4.
Mixing R-sines and modern sines. Divide by R = 120 to compare with your calculator: 31.058/120 = 0.2588 = sin 15°.
For more practice with these identities, see our guide to trigonometric identities and how to practise them.
What can a student learn from Varahamihira?
Varahamihira compared five sources, said openly which he trusted most and why, and then built a working tool from them. In your own study, when two textbooks or teachers explain a topic differently, do the same: write both versions side by side, test each on a problem, and keep the one that works.
When an identity gives a wrong answer and you cannot see why, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Try it now: use Varahamihira's rule to find R sin 7°30′ from R sin 75°, and check it against the table's 15;40.
