Prove the identity of $ sin( heta) $ on the unit circle.

To prove the identity of $ \sin(\theta) $ on the unit circle, we start by considering a point on the unit circle at angle $ \theta $. The coordinates of this point can be represented as $ (\cos(\theta), \sin(\theta)) $.

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Using the Pythagorean identity for the unit circle, we have:

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$ \cos^2(\theta) + \sin^2(\theta) = 1 $

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Now consider a right triangle with the hypotenuse being the radius of the unit circle (which is 1). The opposite side of angle $ \theta $ is $ \sin(\theta) $ and the adjacent side is $ \cos(\theta) $.

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By the definition of sine in a right triangle, we get:

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$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} $

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Since the hypotenuse is 1, the opposite side is $ \sin(\theta) $, thus:

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$ \sin(\theta) = \sin(\theta) $

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This completes the proof.

To derive the value of $ sin( heta) $, consider a point on the unit circle at angle $ heta $. The coordinates of this point are $ (cos( heta), sin( heta)) $.

Using the definition of sine and cosine on the unit circle, we have:

$ cos^2( heta) + sin^2( heta) = 1 $

The value of $ sin( heta) $ is simply the $ y $-coordinate of the point on the unit circle at angle $ heta $. Hence:

$ sin( heta) = sin( heta) $

On the unit circle, $ sin( heta) $ corresponds to the $ y $-coordinate of the point at angle $ heta $. Thus:

$ sin( heta) = sin( heta) $

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