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Before calculators, finding sin 37° meant looking it up in a table or doing a lot of arithmetic. Around 600 CE an Indian astronomer wrote down a single fraction that gives the sine of any angle to about two decimal places using nothing but multiplication and division. This post covers who Bhaskara I was, what his formula says, worked values you can check, how accurate it is, and what you can learn from it for your own trigonometry.
Who was Bhaskara I?
We know little about his life. The MacTutor biography from the University of St Andrews gives his dates as about 600 to about 680 CE and suggests he was born in Saurashtra, in present-day Gujarat, and later worked in a school of followers of Aryabhata at Asmaka. He is called Bhaskara I to distinguish him from the later Bhaskara II, whom NCERT dates to 1114.
He wrote two astronomical treatises, the Mahabhaskariya and the Laghubhaskariya, and in 629 a commentary on Aryabhata's work, the Aryabhatiyabhasya. MacTutor notes that he criticised the common use of √10 for π and believed π was not rational. The historical note in the NCERT Class 11 chapter Trigonometric Functions names "Bhaskara I (600)" among the Indian mathematicians who obtained key results in trigonometry, and says he "gave formulae to find the values of sine functions for angles more than 90°".
What is Bhaskara's sine formula?
MacTutor explains that three verses in chapter 7 of the Mahabhaskariya give the approximation, and that Bhaskara attributes it to Aryabhata. In radians it reads:
sin x ≈ 16x(π − x) / [5π² − 4x(π − x)]
Put x = πθ/180 to convert to degrees. Every π² cancels, and after dividing top and bottom by 4 you get the form that is easier to use by hand:
sin θ ≈ 4θ(180 − θ) / [40500 − θ(180 − θ)]
Notice what the expression θ(180 − θ) does. It is 0 at 0° and at 180°, largest at 90°, and the same for θ and 180° − θ. The real sine has exactly those features, which is a big part of why the fraction fits so well.
Worked example: does it really work?
Let P = θ(180 − θ). Then sin θ ≈ 4P / (40500 − P).
θ = 30°. P = 30 × 150 = 4500. 4P = 18000. 40500 − 4500 = 36000. Result: 18000 ÷ 36000 = 0.5. Exactly right.
θ = 90°. P = 90 × 90 = 8100. 4P = 32400. 40500 − 8100 = 32400. Result: 1. Exactly right.
θ = 45°. P = 45 × 135 = 6075. 4P = 24300. 40500 − 6075 = 34425. Result: 24300 ÷ 34425 = 0.7059. True value: 0.7071. Error: −0.0012.
θ = 60°. P = 60 × 120 = 7200. 4P = 28800. 40500 − 7200 = 33300. Result: 0.8649. True value: 0.8660. Error: −0.0012.
θ = 150°. P = 150 × 30 = 4500, the same as for 30°, so the result is 0.5 again, matching sin 150° = sin 30°.
How accurate is the approximation?
MacTutor computes the formula against the true sine from 0 to π/2 and reports that it "leads to a maximum error of less than one percent". Our own check across every hundredth of a degree from 0° to 180° found the largest absolute error to be about 0.0016, near 11.5°. That is under one percent of the sine's maximum value of 1.
There is one subtlety worth knowing. As a percentage of the true value, the error grows for very small angles. Near 0°, the radian form behaves like 16x/5π, and 16/5π ≈ 1.019, so the formula runs about 1.9% high there. At 15° it gives 0.2604 against a true 0.2588, about 0.6% high.
What mistakes do students make with formulas like this?
Putting degrees into the radian form. The 16x(π − x) version needs x in radians. If you work in degrees, use the 40500 version.
Forgetting the denominator has a minus. It is 40500 − P, not 40500 + P. Check with θ = 90°: you must get exactly 1.
Trusting an approximation everywhere. Always test it at points you know, such as 30° and 90°, and look at where it is weakest.
Using it in an exam answer. In JEE or board exams use exact values and identities. Bhaskara's formula is for understanding and estimation, not for replacing sin 45° = 1/√2.
Trigonometric functions and identities are core to the JEE (Main) 2026 syllabus; check the current year's document. Our post on trigonometry identities practice is a good companion, and you can read about his successor in Bhaskara II and the Lilavati.
What can a student take from how Bhaskara I worked?
Look for the shape first. He matched the sine's zeros, peak and symmetry before worrying about exact values.
Make tools people can use. A fraction anyone could compute by hand was more useful than a perfect but impractical method.
Give credit. He attributed the result to Aryabhata, the teacher of his tradition.
Question accepted values. His doubt about √10 for π shows the same habit of checking.
A practice routine: use the formula for 20°, 40° and 70°, compare each with a calculator, and plot the errors; then explain in one sentence why the error is zero at 30°. If you want to rebuild the unit circle picture of the sine before you try, the Fermi Concept Library covers every concept in mathematics.
Next step: compute sin 18° with Bhaskara's formula and compare it with the exact value (√5 − 1)/4.
