Reviewing precalculus, domains, ranges, other?

**wadaltmon** · September 4th 2012, 06:38 PM

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

**Prove It** · September 4th 2012, 06:51 PM

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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