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Math Help - Stuck finding area under the curve

  1. #1
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    Stuck finding area under the curve

    I'm stuck at the question 51 in my textbook.

    I can find the two equations and know the f(x)=y=1 is greater than g(x). I follow the solutions and get stuck at the part where they go from 1 - cos^2 x = sin^2 x.

    1 - cos^2 x makes sense to me since it's f(x) - g(x) but where does the sin^2 x come from. And from this point on I don't understand why they divide it by 2 and create the integral off the interval.

    One attachment has the textbook question, #51 and the other is my solutions book scan.
    Attached Thumbnails Attached Thumbnails Stuck finding area under the curve-stuck51.png   Stuck finding area under the curve-txt51.png  
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  2. #2
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    Quote Originally Posted by solidstatemath View Post
    I'm stuck at the question 51 in my textbook.

    I can find the two equations and know the f(x)=y=1 is greater than g(x). I follow the solutions and get stuck at the part where they go from 1 - cos^2 x = sin^2 x.

    1 - cos^2 x makes sense to me since it's f(x) - g(x) but where does the sin^2 x come from. And from this point on I don't understand why they divide it by 2 and create the integral off the interval.

    One attachment has the textbook question, #51 and the other is my solutions book scan.
    being a calculus student, you should be familiar with both of these identities ...

    \sin^2{x} + \cos^2{x} = 1

    and ...

    \displaystyle \sin^2{x} = \frac{1-\cos(2x)}{2}
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  3. #3
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    The Pythagorean Identity states that

    \cos^2{x} + \sin^2{x} = 1.

    Thus 1 - \cos^2{x} = \sin^2{x}.

    You should also know the double angle identity

    \cos{2x} = \cos^2{x} - \sin^2{x}.

    A combination of the double angle identity and the Pythagorean Identity gives

    \sin^2{x} = \frac{1 - \cos{2x}}{2}.

    They do this conversion to simplify the problem.
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  4. #4
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    Quote Originally Posted by Prove It View Post
    The Pythagorean Identity states that

    \cos^2{x} + \sin^2{x} = 1.

    Thus 1 - \cos^2{x} = \sin^2{x}.

    They do this conversion to simplify the problem.
    fify
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  5. #5
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    Quote Originally Posted by skeeter View Post
    fify
    What does fify mean?
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  6. #6
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    Quote Originally Posted by Prove It View Post
    What does fify mean?
    fixed it for you
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  7. #7
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    Quote Originally Posted by skeeter View Post
    fixed it for you
    I see, thank you. Serves me right for trying to do this at 4am hahaha.
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  8. #8
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     Sin(x)^2+ Cos(x)^2=1 if you think about it for a sec it is Pythagoras theorem. So  Sin(x)^2+ Cos(x)^2=1 \Rightarrow Sin(x)^2=1-Cos(x)^2. For the next step you can use Sin(\alpha)Sin(\beta)=\frac{1}{2}((-Cos(\alpha + \beta)+Cos(\alpha -\beta))) with alpha=beta and that's it.
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