Prove that the equation $ x^2 + y^2 = 1 $ is satisfied by the coordinates of any point on the unit circle for a given angle $ heta $.
Answer 1
William King
To prove that the equation $ x^2 + y^2 = 1 $ is satisfied by the coordinates of any point on the unit circle for a given angle \theta , we start with the unit circle definition:
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On the unit circle, the coordinates of a point corresponding to an angle $ \theta $ are $ (\cos(\theta), \sin(\theta)) $.
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Consider the equation $ x^2 + y^2 = 1 $.
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Substitute $ x = \cos(\theta) $ and $ y = \sin(\theta) $:
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$ \cos^2(\theta) + \sin^2(\theta) = 1 $
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This identity is known as the Pythagorean identity, and it holds true for all values of $ \theta $. Therefore, the equation $ x^2 + y^2 = 1 $ is satisfied by the coordinates of any point on the unit circle.
Answer 2
Amelia Mitchell
We need to prove that the equation $ x^2 + y^2 = 1 $ holds for coordinates of points on the unit circle for any angle $ heta $.
The coordinates of a point on the unit circle are $ (cos( heta), sin( heta)) $.
Substitute these into the equation:
$ x = cos( heta), y = sin( heta) $
Then,
$ cos^2( heta) + sin^2( heta) = 1 $
This is the Pythagorean identity, proving that $ x^2 + y^2 = 1 $ for any point on the unit circle.
Answer 3
Alex Thompson
For a point on the unit circle:
Coordinates are $ (cos( heta), sin( heta)) $.
Plug into the equation:
$ cos^2( heta) + sin^2( heta) = 1 $
The identity proves $ x^2 + y^2 = 1 $.
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