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Thread: 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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    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 $\displaystyle a = \arcsin\left(\frac{1}{2}\right)$

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

    $\displaystyle b = \arcsin\left(\frac{1}{3}\right)
    $

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


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

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

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

    $\displaystyle a+b = \arcsin\left(\frac{2\sqrt{2} + \sqrt{3}}{6}\right)
    $
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