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Thread: Need help with finding exact value

  1. #1
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    Need help with finding exact value

    Given that q = $\displaystyle sin 1( 3/5)$find the exact value of $\displaystyle sin (1/3π+ q).$
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  2. #2
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    Hello, perryman!

    Given that: .$\displaystyle q \,=\, \sin^{ 1}\left(-\,\frac{3}{5}\right) $

    find the exact value of: .$\displaystyle \sin\left(q + \frac{\pi}{3}\right)$

    We have: .$\displaystyle q \;=\;\sin^{-1}\left(-\,\frac{3}{5}\right) $

    Then: . $\displaystyle \sin q \;=\;-\frac{3}{5} \;=\;\frac{opp}{hyp}$

    $\displaystyle q$ is an angle in a right triangle with: $\displaystyle opp = -3,\;hyp = 5$
    . . Pythagorus says: .$\displaystyle adj = \pm4$
    Hence: .$\displaystyle \cos q \,=\,\pm\frac{4}{5}$


    We have: .$\displaystyle \sin\left(q + \frac{\pi}{3}\right) \;=\;\sin q\cos\frac{\pi}{3} + \cos q\sin\frac{\pi}{3}$

    . . . . . . . . . . . . . $\displaystyle =\;\left(-\frac{3}{5}\right)\left(\frac{1}{2}\right) + \left(\pm\frac{4}{5}\right)\left(\frac{\sqrt{3}}{2 }\right)$

    . . . . . . . . . . . . . $\displaystyle =\;-\frac{3}{10} \pm\frac{4\sqrt{3}}{10}$

    . . . . . . . . . . . . . $\displaystyle =\; \frac{-3 \pm 4\sqrt{3}}{10} $

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