The problem says:
Show if sinθ=√1−cos2θ if sinθ is positive and sinθ=−√1−cos2θ if sinθ is negative.
I know that sin2θ+cos2θ=1 which can be proven with the Pythagorean Theorem. I also understand we can rearrange the above formula to get sinθ=√1−cos2θ, but how to show that if sinθ is negative, then sinθ=−√1−cos2θ?
I can't bring that together with the Pythagorean Theorem (this was my first thought to use) to proof it.
Answer
Note that if x2=16 we get x=±4 where 4 is positive and −4 is negative.
Similarly sin2x=1−cos2x⟹sinx=±√1−cos2x
where the positive sign is for the positive sinx and the negative sign for the negative sinx
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