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Thread: Trigonometric Formulars

  1. #1
    Junior Member
    Joined
    Feb 2006
    Posts
    29

    Trigonometric Formulars

    Have I got this right or have I failed miserably as usual

    Given that $\displaystyle
    0<\theta<\frac{1}{2}pi
    $

    and $\displaystyle
    sin (\theta)= \frac{1}{4}
    $

    using appropriate trigonometric formulas find the exact values of the following
    $\displaystyle
    (i) cos(2\theta)
    $
    Use
    $\displaystyle
    cos(2\theta) = 1-2 sin^2\theta
    $
    gives
    $\displaystyle
    cos(2\theta) = 1-2 * \frac{1}{4}^2 = \frac{7}{8}
    $

    $\displaystyle
    (ii) cos(\theta)
    $
    use
    $\displaystyle
    cos(\theta) = \pm \sqrt{1-sin^2\theta}
    $
    gives
    $\displaystyle
    cos(\theta) = \pm \sqrt{1-\frac{1}{4}^2}= \pm\sqrt{\frac{15}{16}} = \frac{1}{4}\sqrt{15}
    $

    Positive in this case

    $\displaystyle
    (iii) sin(3\theta)
    $

    Rewrite as
    $\displaystyle
    sin(2\theta+\theta)
    $

    find $\displaystyle
    sin(2\theta)
    $ using $\displaystyle
    sin(2\theta) = 2 sin\theta cos \theta
    $

    $\displaystyle
    sin(2\theta) = 2 * \frac {1}{4}*\frac{1}{4}\sqrt{15} = \frac{1}{8}\sqrt{15}
    $


    and then use

    $\displaystyle
    sin(\phi+\theta) = sin\phi cos\theta + cos\phi sin\theta
    $

    Giving

    $\displaystyle
    sin(2\theta+\theta) = \frac{1}{8}\sqrt{15}*\frac{1}{4}\sqrt{15} + \frac{7}{8}*\frac{1}{4} = \frac{11}{16}
    $

    So

    $\displaystyle
    sin(3\theta) = \frac{11}{16}
    $

    Again positive
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  2. #2
    Member
    Joined
    Nov 2005
    From
    Wethersfield, CT
    Posts
    92
    Greetings:

    Your work is well organized and, in the case at hand, 100% correct! Nice work. Now we need only work on that confidence. You are not a "failed miserably as usual" kind of person. You are learning a complex discipline and, from what I can see, you are learning well!

    Regards,

    Rich B.
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  3. #3
    Junior Member
    Joined
    Feb 2006
    Posts
    29
    Quote Originally Posted by Rich B.
    Greetings:

    Your work is well organized and, in the case at hand, 100% correct! Nice work. Now we need only work on that confidence. You are not a "failed miserably as usual" kind of person. You are learning a complex discipline and, from what I can see, you are learning well!

    Regards,

    Rich B.
    Thanks that's given me a big confidence boost !!
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