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Math Help - Find Exact Value of Each Expression

  1. #1
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    Find Exact Value of Each Expression

    Find exact value of each expression.

    (1) sin[2arcsin(sqrt{3}/2)]

    (2) cos^2[1/2arcsin(3/5)]
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  2. #2
    Moo
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    Quote Originally Posted by magentarita View Post
    Find exact value of each expression.

    (1) sin[2arcsin(sqrt{3}/2)]
    \sin(2x)=2 \sin(x) \cos(x)

    Here, x=\arcsin \left(\frac{\sqrt{3}}{2}\right)

    Remember the formula \cos(\arcsin(x))=\sqrt{1-x^2}

    ----> \sin \left(2 \arcsin \left(\frac{\sqrt{3}}{2}\right) \right)=2 \frac{\sqrt{3}}{2} \sqrt{1-\left(\frac{\sqrt{3}}{2}\right)^2}

    etc...


    (2) cos^2[1/2arcsin(3/5)]
    We know that \cos(2x)=2 \cos^2(x)-1 \implies \cos(x)=2 \cos^2 \tfrac x2 -1 \implies \cos^2 \tfrac x2=\tfrac{\cos(x)+1}{2}

    Now, you just have to calculate \cos(\arcsin(\tfrac 35))
    Once again, this formula : \cos(\arcsin(x))=\sqrt{1-x^2}
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  3. #3
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    Fabulous...

    Quote Originally Posted by Moo View Post
    \sin(2x)=2 \sin(x) \cos(x)

    Here, x=\arcsin \left(\frac{\sqrt{3}}{2}\right)

    Remember the formula \cos(\arcsin(x))=\sqrt{1-x^2}

    ----> \sin \left(2 \arcsin \left(\frac{\sqrt{3}}{2}\right) \right)=2 \frac{\sqrt{3}}{2} \sqrt{1-\left(\frac{\sqrt{3}}{2}\right)^2}

    etc...



    We know that \cos(2x)=2 \cos^2(x)-1 \implies \cos(x)=2 \cos^2 \tfrac x2 -1 \implies \cos^2 \tfrac x2=\tfrac{\cos(x)+1}{2}

    Now, you just have to calculate \cos(\arcsin(\tfrac 35))
    Once again, this formula : \cos(\arcsin(x))=\sqrt{1-x^2}
    Fabulous work as always.
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