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Math Help - how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3)/6)

  1. #1
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    how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3)/6)

    hi I NEED help on my A Level maths i am confused on ,
    how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3)/6)

    really appreciate it if you could help

    Sunset
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  2. #2
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    Quote Originally Posted by sunset View Post
    hi I NEED help on my A Level maths i am confused on ,
    how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3^0.5)/6)

    really appreciate it if you could help

    Sunset
    let a = \arcsin\left(\frac{1}{2}\right)

    \sin{a} = \frac{1}{2} ... \cos{a} =  \frac{\sqrt{3}}{2}

    b = \arcsin\left(\frac{1}{3}\right)<br />

    \sin{b} = \frac{1}{3} ... \cos{b} =  \frac{2\sqrt{2}}{3}


    \sin(a+b) = \sin{a}\cos{b} + \cos{a}\sin{b}

    \sin(a+b) = \frac{1}{2} \cdot \frac{2\sqrt{2}}{3} +  \frac{\sqrt{3}}{2} \cdot \frac{1}{3}

    \sin(a+b) = \frac{2\sqrt{2} + \sqrt{3}}{6}

    a+b = \arcsin\left(\frac{2\sqrt{2} + \sqrt{3}}{6}\right)<br />
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