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Math Help - Reviewing precalculus, domains, ranges, other?

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    Reviewing precalculus, domains, ranges, other?

    I am supposed to find the domain and range of some functions, but I don't remember how from last year. I know it has something to do with kπ, and you add it to the function, but I don't really remember what I am supposed to do with it to find the domain and range. What is a general process for doing this? Can you show me how to do this process on the problem "y = 3 csc (3x + π) - 2"?
    Also, it says I have to give the exact values of the six trigonometric functions of something like (theta) = arcsin (8/17). How do I find these?
    Another thing... I need to evaluate expressions like sin(arccos(7/11)). How do I do that?
    Thanks,
    Dalton
    Last edited by wadaltmon; September 4th 2012 at 06:42 PM.
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  2. #2
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    Re: Reviewing precalculus, domains, ranges, other?

    You should be well aware of the Pythagorean Identity: \displaystyle \begin{align*} \sin^2{\theta} + \cos^2{\theta} \equiv 1 \end{align*}.

    From that we get \displaystyle \begin{align*} \sin{\theta} \equiv \pm \sqrt{ 1 - \cos^2{\theta} }  \end{align*}.

    So for \displaystyle \begin{align*} \sin{\left[\arccos{\left(\frac{7}{11}\right)}\right]} \end{align*} we have

    \displaystyle \begin{align*} \sin{\left[\arccos{\left(\frac{7}{11}\right)}\right]} &= \pm \sqrt{ 1 - \left\{ \cos{\left[\arccos{\left(\frac{7}{11}\right)}\right]} \right\}^2 } \\ &= \pm \sqrt{ 1 - \left(\frac{7}{11}\right)^2 } \\ &= \sqrt{ 1 - \frac{49}{121} } \\ &= \sqrt{ \frac{72}{121} } \\ &=  \pm \frac{ \sqrt{72}}{11} \end{align*}
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