Wednesday, January 18, 2017

trigonometry - Describing sine in terms of cosine



The problem says:





Show if sinθ=1cos2θ if sinθ is positive and sinθ=1cos2θ 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=1cos2θ, but how to show that if sinθ is negative, then sinθ=1cos2θ?



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=1cos2xsinx=±1cos2x



where the positive sign is for the positive sinx and the negative sign for the negative sinx


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