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Q. Why is sin(-θ)= - sin(θ)?
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You’re right. With only the right angled triangle definition of sine it doesn’t make sense to think of negative inputs or outputs for the sine function.

You need to use the slightly more “sophisticated” unit circle definition
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You need to think sin(theta) as the height of a point theta degrees around a unit circle (a circle centred at (0,0) and r=1 starting at (1,0)) so you can have positive and negative length and height so for -theta it’d be the point on the opposite side of the x-axis and since sin(theta) is height (or y-value) sin(-theta) would just just be -sin(theta)

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