\documentclass[12pt]{amsart}

\usepackage{amsmath, amssymb, amsthm}
\usepackage[margin=0.5in]{geometry}
\usepackage[pdflang={en-US}]{hyperref}
\usepackage[shortlabels]{enumitem}
\usepackage{multicol}

\usepackage{tikz}

\renewcommand{\thesection}{\Alph{section}}

\newcommand{\scs}[1]{\begin{scshape}#1\end{scshape}}
\newcommand*{\inv}{^{-1}}
\newcommand*{\RR}{\mathbb{R}}
\newcommand*{\QQ}{\mathbb{Q}}
\newcommand*{\ZZ}{\mathbb{Z}}
\newcommand*{\NN}{\mathbb{N}}

\newcommand*{\inte}[2][x]{\int #2\, d#1}
\newcommand*{\der}[2][x]{\frac{d}{d#1}\!\left[ #2 \right]}
\newcommand*{\dery}{\frac{\text{d}y}{\text{d}x}}

\newtheorem*{theorem}{Theorem}
\theoremstyle{definition}
\newtheorem*{definition}{Definition}
\newtheorem{problem}{}
\newtheorem*{challenge}{Challenge}
\newtheorem*{example}{Example}
\newtheorem*{solution}{Solution}
\newtheorem*{ruleofthumb}{Rule of Thumb}

\newcommand{\thm}[1]{\begin{theorem}#1\end{theorem}}
\newcommand{\prob}[1]{\begin{problem}#1\end{problem}}
\newcommand{\dfn}[1]{\begin{definition}#1\end{definition}}
\newcommand{\chal}[1]{\begin{challenge}#1\end{challenge}}
\newcommand{\eg}[1]{\begin{example}#1\end{example}}
\newcommand{\sol}[1]{\begin{solution}#1\end{solution}}
\newcommand{\rot}[1]{\begin{ruleofthumb}#1\end{ruleofthumb}}

\newcommand{\abs}[1]{\left| #1 \right|}

\title{Inverse trig functions}
\date{October 11, 2021}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\prob{
Evaluate the following:
\begin{enumerate}[a)]
\item $\displaystyle \sin^{-1}\left( -\frac{\sqrt{3}}{2} \right)$

\vfill

\item $\displaystyle \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right)$

\vfill

\item $\displaystyle \tan^{-1}\left( 1 \right)$

\vfill

\item $\displaystyle \sin^{-1}\left( \sin\left( -\frac{\pi}{6} \right) \right)$

\vfill

\item $\displaystyle \sin^{-1}\left( \sin\left( \frac{2\pi}{3} \right) \right)$

\vfill

\item $\displaystyle \tan^{-1}\left( \tan\left( \pi \right) \right)$

\vfill

\end{enumerate}
}

\newpage

\prob{
Use implicit differentiation and the triangle trick from last class to find $\displaystyle \der{\cos^{-1}x}$.

\vfill

}


\prob{
Find the following derivatives.
\begin{enumerate}[a)]
\item $\displaystyle \der{\cos^{-1}\left( \frac{1}{x} \right)}$

\vfill

\item $\displaystyle \der{\tan^{-1}\left( x^2 \right)}$

\vfill

\item $\displaystyle \der{x \sin^{-1}\left( 5x \right)}$

\vfill

\item $\displaystyle \der{\tan\left( \cos^{-1}\left( \frac{x}{5} \right) \right)}$

\vfill

\end{enumerate}
}

\chal{
Use implicit differentiation and the triangle trick from last class to find $\displaystyle \der{\sec^{-1}x}$.
}

\end{document}