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Math Help - sin and cos

  1. #1
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    sin and cos

    When \frac{\pi}{2}<a<\pi, sina=\frac{4}{5}

    cosa\:=

    not quite sure how to go from sin to cos... not quite sure what to do...

    the second part of the question is

    sin2a\:=

    cos2a\:=
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  2. #2
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    SinA=\frac{opposite}{hypotenuse}

    and from Pythagoras' theorem

    adjacent=\sqrt{(hypotenuse)^2-(opposite)^2}
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  3. #3
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    Ok,

    adjacent=\sqrt{(hypotenuse)^2-(opposite)^2}=3

    cosA=\frac{adjacent}{hypotenuse}=\frac{3}{5}

    so far so good, I guess... but the book says the answer is -\frac{3}{5}
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  4. #4
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    This is because of the range given,

    90^o<a<180^o

    Cosine locates the x co-ordinate of a point on the circle circumference.
    Once you have discovered the "distance" the point is from the origin, you simply invert it in this case as the point lies to the left of the y-axis..

    Cos(A)=-Cos(180-A)
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  5. #5
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    here is a sketch
    Attached Thumbnails Attached Thumbnails sin and cos-rcosa.jpg  
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  6. #6
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    Ok, thanks... googled around a bit and found doubleangle formulae...

    sin2a=2sin\:a\:cos\:a\:\:,\:\:cos2a=cos^2a-sin^2a

    and I'm pretty sure this made an appearance in my earlier thread now that I think of it...

    thus;

    sin2a=2sin\:a\:cos\:a=2\times{\frac{4}{5}}\times{-\frac{3}{5}}=-\frac{24}{25}

    cos2a=cos^2a-sin^2a=(-\frac{3}{5})^2-(\frac{4}{5})^2=-\frac{7}{25}

    Many thanks!
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  7. #7
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    That's the idea,
    you find the answers of the "double angle" situation,
    using your results with the "single angle".

    Good work!
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