pi is the ratio between the circumference of a circle and it's diameter
sine is the ratio between the side opposite an angle of a right triangle and it's hypotenuse
both depend on the IDEA of the ratio - so once someone thought of the possibility of expressing a relationship as a ratio/fraction, all the tools needed to express pi and sine were there.
then you need someone to apply this idea to some parts of a circle and some parts of a triangle - so the real question is "to which parts of which shape did someone first try and find ratios" and to this I have no answer.
if you are wondering which was calculated first, again i have no idea - it depends on which shape our hypothetical mathematician decided to fiddle with first. there are some relatively easy sines:
30-60-90 triangles have a side which is half the length of the hypotenuse - this ratio is easy to recognize, so someone may have thought about this idea.
if you're asking which was defined by humans first, i'd probably guess pi, but this is a question for a historian really, and it doesn't actually matter. giving a name to the concept doesn't make it exist. there were prime numbers long before anyone realized that they existed, and there were irrational numbers long before anyone thought of taking square roots.
as long as there have been circles there has been pi = 3.14159... and as long as there have been triangles there have been the sines of angles.
and frankly, there have been circles as long as triangles have been capable of having hypotenuses of the same length.
sine is the ratio between the side opposite an angle of a right triangle and it's hypotenuse
both depend on the IDEA of the ratio - so once someone thought of the possibility of expressing a relationship as a ratio/fraction, all the tools needed to express pi and sine were there.
then you need someone to apply this idea to some parts of a circle and some parts of a triangle - so the real question is "to which parts of which shape did someone first try and find ratios" and to this I have no answer.
if you are wondering which was calculated first, again i have no idea - it depends on which shape our hypothetical mathematician decided to fiddle with first. there are some relatively easy sines:
30-60-90 triangles have a side which is half the length of the hypotenuse - this ratio is easy to recognize, so someone may have thought about this idea.
if you're asking which was defined by humans first, i'd probably guess pi, but this is a question for a historian really, and it doesn't actually matter. giving a name to the concept doesn't make it exist. there were prime numbers long before anyone realized that they existed, and there were irrational numbers long before anyone thought of taking square roots.
as long as there have been circles there has been pi = 3.14159... and as long as there have been triangles there have been the sines of angles.
and frankly, there have been circles as long as triangles have been capable of having hypotenuses of the same length.
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I guess pi because the ancient Greeks understood areas and circumferences. Sine is more artificial.
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apple pi before hazard sign