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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Practice on Trigonometric Functions}
\centerline{September 13, 2017}
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\noindent
{\it Background}
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In the video and class yesterday, we have seen a ``lightning review'' of trigonometry.
To practice on some ideas related to this, do the following problems.
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)] Sinusoids
\begin{enumerate}
\item[(a)] Starting from the graph $y = \sin(x)$ and using ideas we have seen about scaling and shifting, 
sketch the graph 
$$y = 3 \sin\left(\frac{x}{2}\right) - 1$$ 
on the interval $[0,8\pi]$.  
\item[(b)] The graph in part (a) is an example of a \emph{sinusoid} (or sine-wave graph).  The 
\emph{amplitude} of a sinusoid is one-half the vertical distance between the minimum and 
maximum values.  What is the amplitude of your sinusoid in (a)?
\item[(c)] The \emph{period} of a sinusoid $f(x)$ is the {\it smallest strictly positive number} $T$
for which it is true that $f(x + T) = f(x)$ for all $x$.  
For example the period of $f(x) = \cos(x)$ is $T = 2\pi$.
What is the period of the sinusoid from part (a)?
\item[(d)]  Repeat parts (a), (b), and (c) for the graph 
$$y = -\cos(3x) + 2$$
on the interval $[-4\pi/3, 4\pi/3]$.  
\item[(e)]  Give a formula defining a sinusoid with amplitude $A = 7$ and period $T = 5\pi$.
(There is more than one correct answer!)  
\end{enumerate}
\item[2)]  By plotting points with $0 < x < \pi$, sketch the graph $y = \cot(x) = \frac{\cos(x)}{\sin(x)}$.
How are $\tan\left(x-\frac{\pi}{2}\right)$ and $\cot(x)$ related?  
\end{enumerate}
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