Skip to main content Contents Prev Up Next \(\require{cancel}\newcommand{\dee}[1]{\mathrm{d}#1}
\newcommand{\half}{ \frac{1}{2} }
\newcommand{\ds}{\displaystyle}
\newcommand{\ts}{\textstyle}
\newcommand{\es}{ {\varnothing}}
\newcommand{\st}{ {\mbox{ s.t. }} }
\newcommand{\pow}[1]{ \mathcal{P}\left(#1\right) }
\newcommand{\set}[1]{ \left\{#1\right\} }
\newcommand{\lin}{{\text{LIN}}}
\newcommand{\quot}{{\text{QR}}}
\newcommand{\simp}{{\text{SMP}}}
\newcommand{\diff}[2]{ \frac{\mathrm{d}#1}{\mathrm{d}#2}}
\newcommand{\bdiff}[2]{ \frac{\mathrm{d}}{\mathrm{d}#2} \left( #1 \right)}
\newcommand{\ddiff}[3]{ \frac{\mathrm{d}^#1#2}{\mathrm{d}{#3}^#1}}
\renewcommand{\neg}{ {\sim} }
\newcommand{\limp}{ {\;\Rightarrow\;} }
\newcommand{\nimp}{ {\;\not\Rightarrow\;} }
\newcommand{\liff}{ {\;\Leftrightarrow\;} }
\newcommand{\niff}{ {\;\not\Leftrightarrow\;} }
\newcommand{\De}{\Delta}
\newcommand{\bbbn}{\mathbb{N}}
\newcommand{\bbbr}{\mathbb{R}}
\newcommand{\bbbp}{\mathbb{P}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\Si}{\Sigma}
\newcommand{\arccsc}{\mathop{\mathrm{arccsc}}}
\newcommand{\arcsec}{\mathop{\mathrm{arcsec}}}
\newcommand{\arccot}{\mathop{\mathrm{arccot}}}
\newcommand{\erf}{\mathop{\mathrm{erf}}}
\newcommand{\smsum}{\mathop{{\ts \sum}}}
\newcommand{\atp}[2]{ \genfrac{}{}{0in}{}{#1}{#2} }
\newcommand{\ave}{\mathrm{ave}}
\newcommand{\llt}{\left \lt }
\newcommand{\rgt}{\right \gt }
\newcommand{\YEaxis}[2]{\draw[help lines] (-#1,0)--(#1,0) node[right]{$x$};\draw[help lines] (0,-#2)--(0,#2) node[above]{$y$};}
\newcommand{\YEaaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$x$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};}
\newcommand{\YEtaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$t$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};}
\newcommand{\YEtaaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$t$}; \draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};}
\newcommand{\YExcoord}[2]{\draw (#1,.2)--(#1,-.2) node[below]{$#2$};}
\newcommand{\YEycoord}[2]{\draw (.2,#1)--(-.2,#1) node[left]{$#2$};}
\newcommand{\YEnxcoord}[2]{\draw (#1,-.2)--(#1,.2) node[above]{$#2$};}
\newcommand{\YEnycoord}[2]{\draw (-.2,#1)--(.2,#1) node[right]{$#2$};}
\newcommand{\YEstickfig}[3]{
\draw (#1,#2) arc(-90:270:2mm);
\draw (#1,#2)--(#1,#2-.5) (#1-.25,#2-.75)--(#1,#2-.5)--(#1+.25,#2-.75) (#1-.2,#2-.2)--(#1+.2,#2-.2);}
\newcommand{\IBP}[7]{
\begin{array}{|c | l | l |}
\hline
\color{red}{\text{Option 1:}}
& u=#2
&\color{red}{\dee{u}=#3 ~ \dee{#1}}
\\
& \dee{v}=#5~\dee{#1}
&\color{red}{v=#7}
\\
\hline
\color{blue}{\text{Option 2:}}
& u=#5
&\color{blue}{\dee{u}=#6 ~ \dee{#1}}
\\
&\dee{v}=#2 \dee{#1}
&\color{blue}{v=#4}
\\
\hline
\end{array}
}
\renewcommand{\textcolor}[2]{{\color{#1}{#2}}}
\newcommand{\trigtri}[4]{
\begin{tikzpicture}
\draw (-.5,0)--(2,0)--(2,1.5)--cycle;
\draw (1.8,0) |- (2,.2);
\draw[double] (0,0) arc(0:30:.5cm);
\draw (0,.2) node[right]{$#1$};
\draw (1,-.5) node{$#2$};
\draw (2,.75) node[right]{$#3$};
\draw (.6,1.1) node[rotate=30]{$#4$};
\end{tikzpicture}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section A.9 Inverse Trigonometric Functions
Some of you may not have studied inverse trigonometric functions in highschool, however we still expect you to know them by the end of the course.
\begin{equation*}
\arcsin x
\end{equation*}
\begin{equation*}
\arccos x
\end{equation*}
\begin{equation*}
\arctan x
\end{equation*}
Domain: \(-1 \leq x \leq 1\)
Domain: \(-1 \leq x \leq 1\)
Range: \(-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}\)
Range: \(0 \leq \arccos x \leq \pi\)
Range: \(-\frac{\pi}{2} \lt \arctan x \lt \frac{\pi}{2}\)
Since these functions are inverses of each other we have
\begin{align*}
\arcsin(\sin \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\\
\arccos(\cos \theta) &= \theta & 0 \leq \theta \leq \pi\\
\arctan(\tan \theta) &= \theta & -\frac{\pi}{2} \lt \theta \lt \frac{\pi}{2}
\end{align*}
and also
\begin{align*}
\sin(\arcsin x) &= x & -1 \leq x \leq 1\\
\cos(\arccos x) &= x & -1 \leq x \leq 1\\
\tan(\arctan x) &= x & \text{any real } x
\end{align*}
\begin{equation*}
\arccsc x
\end{equation*}
\begin{equation*}
\arcsec x
\end{equation*}
\begin{equation*}
\arccot x
\end{equation*}
Range: \(-\frac{\pi}{2} \leq \arccsc x \leq \frac{\pi}{2}\)
Range: \(0 \leq \arcsec x \leq \pi\)
Range: \(0 \lt \arccot x \lt \pi\)
\begin{equation*}
\arccsc x \ne 0
\end{equation*}
\begin{equation*}
\arcsec x \ne \frac{\pi}{2}
\end{equation*}
Again
\begin{align*}
\arccsc(\csc \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2},\ \theta\ne 0\\
\arcsec(\sec \theta) & = \theta & 0 \leq \theta \leq \pi,\ \theta\ne \frac{\pi}{2}\\
\arccot(\cot \theta) & = \theta & 0 \lt \theta \lt \pi
\end{align*}
and
\begin{align*}
\csc(\arccsc x) &= x & |x|\ge 1\\
\sec(\arcsec x) &= x & |x|\ge 1\\
\cot(\arccot x) &= x & \text{any real } x
\end{align*}
