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The radians provide absolute no help for your problem. There are alternative trigonometric functions with the argument measured in cycles that can be used inst

The radians provide absolute no help for your problem. There are alternative trigonometric functions with the argument measured in cycles that can be used instead of those with argument measured in radians. All formulae written with sin, cos and atan can be rewritten with sin2pi, cos2pi and atan2pi, or whatever names you want for those functions. Moreover, before the electronic computers, nobody in the entire history of mankind had used trigonometric functions with arguments measured in radians for any practical problem, like astronomical or geodesy problems. The radians became popular in the 19th century, only for symbolic computations, because they simplify the formulae for derivatives and primitives. For all numeric computations, even in the 19th century, nobody used trigonometric functions with arguments measured in radians, but only trigonometric functions with arguments measured in sexagesimal degrees, which behave much more similarly to the functions with arguments measured in cycles, than to the functions with arguments measured in radians, which cannot be computed exactly for any rational argument, and where the argument can never be reduced exactly to a value in the first quadrant. While the trigonometric functions with arguments in radians are harmful, the constant 2Pi remains an essential constant, which must be used to convert between the length and the central angle that correspond to a circular arc, and also to compute the derivatives and primitives of the exponential 1^x and of the trigonometric functions.

srean 2mo ago
Some anecdotes to support and color your point. The sin function has Indian and Greek roots. Part of that history is in the name 'sin' itself. I will let you all look that up. It's quite a funny story especially the jya to jayb bit. This comment is about computation of sin in Indian mathematics and its definition. The analogue of the sin function in Indian mathematics was a function from length to length. It was defined as the length of the half chord that corresponds to the angle, not of a unit radius circle but of a circle of radius 3438 units. But why 3438 ? That requires understanding the Indian system of measuring angles by length. They wanted to have a numeric precision of 1 arc minute but weren't very fond of manipulating fractions. So they incorporated their requirement in how they standardized their circle -- Not by radius or diameter but by arc length. They chose that circle to be the standard for which the arc length of one arc minute is unit length. If one does the math, it is 360 * 60 / 2π. Plugin the accuracy of π that was known to Indian mathematicians of that time you get a radius of 3438 units. If you are curious about their numeric calculation of sin look up Aryabhatta and Bhaskara. Wikipedia is quite informative. https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa's_sine_table https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa's_sine_... https://en.wikipedia.org/wiki/Bh%C4%81skara_I%27s_sine_approximation_formula https://en.wikipedia.org/wiki/Bh%C4%81skara_I%27s_sine_appro...
jacobolus 2mo ago
> The radians became popular in the 19th century Something akin to radians was widely used in ancient India. That is, sine tables constructed for a circle with circumference 360·60 (angular minutes) and radius 3438 ≈ 360·60/2π. In Europe, radians per se developed in the early 18th century. What is true is that the name radian came about in the late 19th century. > trigonometric functions with arguments in radians are harmful Trigonometric functions are not inherently "harmful", but they are sometimes cumbersome. Changing the units doesn't make all that much practical difference. Avoiding transcendental functions in favor of vector methods is often a good strategy though, especially in code.
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srean 2mo ago

Some anecdotes to support and color your point. The sin function has Indian and Greek roots. Part of that history is in the name 'sin' itself. I will let you all look that up. It's quite a funny story especially the jya to jayb bit. This comment is about computation of sin in Indian mathematics and its definition. The analogue of the sin function in Indian mathematics was a function from length to length. It was defined as the length of the half chord that corresponds to the angle, not of a unit radius circle but of a circle of radius 3438 units. But why 3438 ? That requires understanding the Indian system of measuring angles by length. They wanted to have a numeric precision of 1 arc minute but weren't very fond of manipulating fractions. So they incorporated their requirement in how they standardized their circle -- Not by radius or diameter but by arc length. They chose that circle to be the standard for which the arc length of one arc minute is unit length. If one does the math, it is 360 * 60 / 2π. Plugin the accuracy of π that was known to Indian mathematicians of that time you get a radius of 3438 units. If you are curious about their numeric calculation of sin look up Aryabhatta and Bhaskara. Wikipedia is quite informative. https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa's_sine_table https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa's_sine_... https://en.wikipedia.org/wiki/Bh%C4%81skara_I%27s_sine_approximation_formula https://en.wikipedia.org/wiki/Bh%C4%81skara_I%27s_sine_appro...

jacobolus 2mo ago

> The radians became popular in the 19th century Something akin to radians was widely used in ancient India. That is, sine tables constructed for a circle with circumference 360·60 (angular minutes) and radius 3438 ≈ 360·60/2π. In Europe, radians per se developed in the early 18th century. What is true is that the name radian came about in the late 19th century. > trigonometric functions with arguments in radians are harmful Trigonometric functions are not inherently "harmful", but they are sometimes cumbersome. Changing the units doesn't make all that much practical difference. Avoiding transcendental functions in favor of vector methods is often a good strategy though, especially in code.

deleted 2mo ago
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