How was it determined that a parsec is light years?

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Parallax is an effective way to measure distance to nearby stars because it relies on geometry. When astronomers use parallax, they are measuring how a star appears to move against its background. The unit parsec refers to the distance that an object would have to be from the Earth  to have a parallax angle of arc-second.
In the figure above, the angle, is the angle that is measured by the Earth on opposite sides of the sun. The parallax angle, is half of this angle.
If we defines to be arc-second, then our object will be parsec away. Since light from the sun takes minutes and seconds to reach the Earth,. we know that;
light minutes
We can use this information to convert our parsec into light years with the tangent formula.
If we convert   to radians, than we can use the small angle approximation, .
Plugging this in for   and using the small angle approximation;
= light-minutes
= light years

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