1018
  • 1018
find the exact value of sin^2(1) + ... + sin^2(90)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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1018
  • 1018
oh, ok. haha. i have some questions after. And also, something's up with my OpenStudy, I can't see your previous replies anymore. Or did you do that? Haha
anonymous
  • anonymous
Yeah, I deleted them. No sense in keeping a wrong answer up.
1018
  • 1018
Oh, I thought my connection's acting weird again. Haha. Thanks.

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anonymous
  • anonymous
Another mistake... \[\sum_{n=1}^{90}\sin^2n^\circ=\frac{1}{2}\sum_{n=0}^{90}(1-\cos2n^\circ)=\color{red}{\frac{91}{2}}-\frac{1}{2}\sum_{n=0}^{90}(\cos^2n^\circ-\sin^2n^\circ)\]
1018
  • 1018
is this ok? Haha, I saw you still typing earlier so I thought you weren't done.
anonymous
  • anonymous
It's a work in progress. You can take advantage of symmetry to reduce this sum nicely, but I forget the details of that approach. What I'm currently doing is taking me in circles...
1018
  • 1018
Hey thanks anyway, maybe I'll just skip this for now. I have a lot of Math to cover haha in so little time. Appreciate the help! I got a lot of questions still, from algebra-calculus, if you still have the time to help later, it would be much appreciated! :)
anonymous
  • anonymous
hi
1018
  • 1018
hello
anonymous
  • anonymous
You can see the symmetry in the cosine sum: \[\frac{91}{2}-\frac{1}{2}\sum_{n=0}^{90}\cos2n^\circ\]with \[\color{red}{\cos0^\circ}+\color{green}{\cos2^\circ}+\color{blue}{\cos4^\circ}+\cdots+\color{blue}{\cos176^\circ}+\color{green}{\cos178^\circ}+\color{red}{\cos180^\circ}\]This reduces to zero, since \(\cos x^\circ=-\cos(180-x)^\circ\).

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