From 30b2c80b12f81ebba914334f500e199b3c3aaa89 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Thu, 2 Aug 2018 19:05:46 +1000 Subject: [PATCH] render final notes for graphing & circ fn's, add local graphics --- methods/circ-functions.html | 1147 +++++++++++++++++++++++++++ methods/circ-functions.md | 75 +- methods/circ-functions.pdf | Bin 0 -> 165823 bytes methods/circ-functions.tex | 215 +++++ methods/exact-values-1.png | Bin 0 -> 5475 bytes methods/exact-values-2.png | Bin 0 -> 6260 bytes methods/graphics/exact-values-1.png | Bin 0 -> 8111 bytes methods/graphics/exact-values-2.png | Bin 0 -> 6445 bytes physics/midyear.fdb_latexmk | 6 +- spec/graphics/recip-parabola.png | Bin 0 -> 105456 bytes spec/graphics/recip-sin-cos.png | Bin 0 -> 235113 bytes spec/graphics/rose.png | Bin 0 -> 119438 bytes spec/graphing.md | 30 +- spec/graphing.pdf | Bin 0 -> 398169 bytes 14 files changed, 1441 insertions(+), 32 deletions(-) create mode 100644 methods/circ-functions.html create mode 100644 methods/circ-functions.pdf create mode 100644 methods/circ-functions.tex create mode 100644 methods/exact-values-1.png create mode 100644 methods/exact-values-2.png create mode 100644 methods/graphics/exact-values-1.png create mode 100644 methods/graphics/exact-values-2.png create mode 100644 spec/graphics/recip-parabola.png create mode 100644 spec/graphics/recip-sin-cos.png create mode 100644 spec/graphics/rose.png create mode 100644 spec/graphing.pdf diff --git a/methods/circ-functions.html b/methods/circ-functions.html new file mode 100644 index 0000000..2b96c4f --- /dev/null +++ b/methods/circ-functions.html @@ -0,0 +1,1147 @@ + + + + + circ-functions + + + + + +

Circular functions

+

Radians and degrees

+ +

Exact values

+

and graphs

+ + +

where
+ is the -dilation (amplitude)
+ is the -dilation (period)
+ is the -shift (phase)
+ is the -shift (equilibrium position)

+

Domain is
+Range is ;

+

Graph of starts at . Graph of starts at .

+

Mean / equilibrium: line that the graph oscillates around ()

+

Solving trig equations

+
    +
  1. Solve domain for
  2. +
  3. Find solutions for
  4. +
  5. Divide solutions by
  6. +
+


+
+
+

+

Amplitude

+

Amplitude of means graph oscillates between and in -axis

+

produces straight line
+ inverts the phase ( becomes , vice vera)

+

Period

+

Period is
+ produces straight line
+ inverts the phase

+

Phase

+

moves the graph left-right in the axis.
+If , the graph has no actual phase shift.

+

Symmetry

+ + + + +

Pythagorean identity

+ +

Complementary relationships

+ + + + +

graph

+ +

where
+ is -dilation (period)
+ is -dilation ( amplitude)
+period is
+range is
+roots at
+asymptotes at
+Asymptotes should always have equations and arrow pointing up

+ + + diff --git a/methods/circ-functions.md b/methods/circ-functions.md index f30ed7a..e639e70 100644 --- a/methods/circ-functions.md +++ b/methods/circ-functions.md @@ -1,12 +1,21 @@ +--- +geometry: margin=2cm +columns: 2 +graphics: yes +--- # Circular functions -## Radians and degrees + -$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$ + ## Exact values +\includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png} +\includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png} + + ## $\sin$ and $\cos$ graphs @@ -14,45 +23,41 @@ $$f(x)=a \sin(bx-c)+d$$ $$f(x)=a \cos(bx-c)+d$$ where -$a$ is the $y$-dilation (amplitude) -$b$ is the $x$-dilation (period) -$c$ is the $x$-shift (phase) -$d$ is the $y$-shift (equilibrium position) + +- $a$ is the $y$-dilation (amplitude) +- $b$ is the $x$-dilation (period) +- $c$ is the $x$-shift (phase) +- $d$ is the $y$-shift (equilibrium position) + Domain is $\mathbb{R}$ + Range is $[-b+c, b+c]$; Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$. **Mean / equilibrium:** line that the graph oscillates around ($y=d$) -## Solving trig equations - -1. Solve domain for $n\theta$ -2. Find solutions for $n\theta$ -3. Divide solutions by $n$ - -$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$ -$2\theta=\sin^{-1}{\sqrt{3} \over 2}$ -$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$ -$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$ - ### Amplitude -Amplitude of $a$ means graph oscillates between $+a$ and $-a$ in $y$-axis +Graph oscillates between $+a$ and $-a$ in $y$-axis $a=0$ produces straight line -$a\lt0$ inverts the phase ($\sin$ becomes $\cos$, vice vera) + +$a < 0$ inverts the phase ($\sin$ becomes $\cos$, vice vera) ### Period Period $T$ is ${2 \pi}\over b$ + $b=0$ produces straight line -$b\lt0$ inverts the phase + +$b<0$ inverts the phase ### Phase $c$ moves the graph left-right in the $x$ axis. + If $c=T={{2\pi}\over b}$, the graph has no actual phase shift. ## Symmetry @@ -75,15 +80,31 @@ $$\cos({\pi \over 2} - \theta)=\sin\theta$$ $$\sin\theta=-\cos(\theta+{\pi \over 2})$$ $$\cos\theta=\sin(\theta+{\pi \over 2})$$ -## $tan$ graph +## $\tan$ graph $$y=a\tan(nx)$$ where -$a$ is $x$-dilation (period) -$n$ is $y$-dilation ($\equiv$ amplitude) -period $T$ is $\pi \over n$ -range is $R$ -roots at $x={k\pi \over n}$ -asymptotes at $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$ + +- $a$ is $x$-dilation (period) +- $n$ is $y$-dilation ($\equiv$ amplitude) +- period $T$ is $\pi \over n$ +- range is $R$ +- roots at $x={k\pi \over n}$ +- asymptotes at $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$ + **Asymptotes should always have equations and arrow pointing up** + +## Solving trig equations + +1. Solve domain for $n\theta$ +2. Find solutions for $n\theta$ +3. Divide solutions by $n$ + +$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$ + +$2\theta=\sin^{-1}{\sqrt{3} \over 2}$ + +$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$ + +$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$ diff --git a/methods/circ-functions.pdf b/methods/circ-functions.pdf new file mode 100644 index 0000000000000000000000000000000000000000..84f3121b4646c395aa13270f7608f013ed928ae3 GIT binary patch literal 165823 zcmb@t1yCKq*DrW+cXxt@;O_439^9Sa1PugtcXxLPPH=a34-O%??(qNKw{NR!A3wh{mr_Ygf}WXy9iDP}eqb4%otc!0)WO&qo{tZnQO?ZX!o`x5m6eT) z^gkbXMhPoh7c(bPMhROZ7c((469-c>cmV-;XBQ_kBRhDH)d^i`2Lf*Fu4m18Wr-Nv zi!sxu>C;>S+}xB*ssz!Mc)PqNuK3J@t$WGEQFNQQGsr+t99`XTOZO$efY1Bk$--2o zxg7c@_NmQvGZ~C?G4o%&k}O^g!<&=7TIUWqi;r*L86wj+`0X5GDmLyMUoR$V2KFMk zihB3hG7w~3Zub>)uGdWdE{E7mSUf&i;bpf!9Zz8OS@V4*Dyt-0tvz#dzI_XaHBUrB z&~fO~3>dkBVKn%=Q(?eC_x#QMmiJA)o#hK`Qswt>SYgjrgWKxm>X$&>xz%X*mJHT| 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1\fi\ifluatex 1\fi=0 % if pdftex + \usepackage[T1]{fontenc} + \usepackage[utf8]{inputenc} + \usepackage{textcomp} % provides euro and other symbols +\else % if luatex or xelatex + \usepackage{unicode-math} + \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} +\fi +% use upquote if available, for straight quotes in verbatim environments +\IfFileExists{upquote.sty}{\usepackage{upquote}}{} +% use microtype if available +\IfFileExists{microtype.sty}{% +\usepackage[]{microtype} +\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts +}{} +\IfFileExists{parskip.sty}{% +\usepackage{parskip} +}{% else +\setlength{\parindent}{0pt} +\setlength{\parskip}{6pt plus 2pt minus 1pt} +} +\usepackage{hyperref} +\hypersetup{ + pdfborder={0 0 0}, + breaklinks=true} +\urlstyle{same} % don't use monospace font for urls +\usepackage[margin=2cm]{geometry} +\usepackage{multicol} +\newcommand{\columnsbegin}{\begin{multicols}{2}} +\newcommand{\columnsend}{\end{multicols}} +\setlength\columnsep{20pt} +\usepackage{graphicx,grffile} +\makeatletter +\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} +\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} +\makeatother +% Scale images if necessary, so that they will not overflow the page +% margins by default, and it is still possible to overwrite the defaults +% using explicit options in \includegraphics[width, height, ...]{} +\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} +\setlength{\emergencystretch}{3em} % prevent overfull lines +\providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} +\setcounter{secnumdepth}{0} +% Redefines (sub)paragraphs to behave more like sections +\ifx\paragraph\undefined\else +\let\oldparagraph\paragraph +\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} +\fi +\ifx\subparagraph\undefined\else +\let\oldsubparagraph\subparagraph +\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} +\fi + +% set default figure placement to htbp +\makeatletter +\def\fps@figure{htbp} +\makeatother + + +\date{} + +\begin{document} + +\columnsbegin +\hypertarget{circular-functions}{% +\section{Circular functions}\label{circular-functions}} + +\hypertarget{radians-and-degrees}{% +\subsection{Radians and degrees}\label{radians-and-degrees}} + +\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\] + +\hypertarget{exact-values}{% +\subsection{Exact values}\label{exact-values}} + +\includegraphics[scale=0.5]{./graphics/exact-values-1.png} + +\hypertarget{sin-and-cos-graphs}{% +\subsection{\texorpdfstring{\(\sin\) and \(\cos\) +graphs}{\textbackslash{}sin and \textbackslash{}cos graphs}}\label{sin-and-cos-graphs}} + +\[f(x)=a \sin(bx-c)+d\] \[f(x)=a \cos(bx-c)+d\] + +where + +\begin{itemize} +\tightlist +\item + \(a\) is the \(y\)-dilation (amplitude) +\item + \(b\) is the \(x\)-dilation (period) +\item + \(c\) is the \(x\)-shift (phase) +\item + \(d\) is the \(y\)-shift (equilibrium position) +\end{itemize} + +Domain is \(\mathbb{R}\) + +Range is \([-b+c, b+c]\); + +Graph of \(\cos(x)\) starts at \((0,1)\). Graph of \(\sin(x)\) starts at +\((0,0)\). + +\textbf{Mean / equilibrium:} line that the graph oscillates around +(\(y=d\)) + +\hypertarget{amplitude}{% +\subsubsection{Amplitude}\label{amplitude}} + +Amplitude of \(a\) means graph oscillates between \(+a\) and \(-a\) in +\(y\)-axis + +\(a=0\) produces straight line + +\(a < 0\) inverts the phase (\(\sin\) becomes \(\cos\), vice vera) + +\hypertarget{period}{% +\subsubsection{Period}\label{period}} + +Period \(T\) is \({2 \pi}\over b\) + +\(b=0\) produces straight line + +\(b<0\) inverts the phase + +\hypertarget{phase}{% +\subsubsection{Phase}\label{phase}} + +\(c\) moves the graph left-right in the \(x\) axis. + +If \(c=T={{2\pi}\over b}\), the graph has no actual phase shift. + +\hypertarget{symmetry}{% +\subsection{Symmetry}\label{symmetry}} + +\[\sin(\theta+{\pi\over 2})=\sin\theta\] +\[\sin(\theta+\pi)=-\sin\theta\] + +\[\cos(\theta+{\pi \over 2})=-\cos\theta\] +\[\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)\] + +\hypertarget{pythagorean-identity}{% +\subsection{Pythagorean identity}\label{pythagorean-identity}} + +\[\cos^2\theta+\sin^2\theta=1\] + +\hypertarget{complementary-relationships}{% +\subsection{Complementary +relationships}\label{complementary-relationships}} + +\[\sin({\pi \over 2} - \theta)=\cos\theta\] +\[\cos({\pi \over 2} - \theta)=\sin\theta\] + +\[\sin\theta=-\cos(\theta+{\pi \over 2})\] +\[\cos\theta=\sin(\theta+{\pi \over 2})\] + +\hypertarget{tan-graph}{% +\subsection{\texorpdfstring{\(\tan\) +graph}{\textbackslash{}tan graph}}\label{tan-graph}} + +\[y=a\tan(nx)\] + +where + +\begin{itemize} +\tightlist +\item + \(a\) is \(x\)-dilation (period) +\item + \(n\) is \(y\)-dilation (\(\equiv\) amplitude) +\item + period \(T\) is \(\pi \over n\) +\item + range is \(R\) +\item + roots at \(x={k\pi \over n}\) +\item + asymptotes at \(x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}\) +\end{itemize} + +\textbf{Asymptotes should always have equations and arrow pointing up} + +\hypertarget{solving-trig-equations}{% +\subsection{Solving trig equations}\label{solving-trig-equations}} + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\tightlist +\item + Solve domain for \(n\theta\) +\item + Find solutions for \(n\theta\) +\item + Divide solutions by \(n\) +\end{enumerate} + +\(\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])\) + +\(2\theta=\sin^{-1}{\sqrt{3} \over 2}\) + +\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\) + +\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\) +\columnsend +\end{document} diff --git a/methods/exact-values-1.png b/methods/exact-values-1.png new file mode 100644 index 0000000000000000000000000000000000000000..4fde496d217fab102627ff57c4f1a1637fe1d66c GIT binary patch literal 5475 zcmeHLdpOf=+}~z0LUKsX%Q16IOyn?!FigxwPDzfVnV4hA$RU)3Axnr2qnDWdo1d7k%q-}k@w@ArClUAuN4zUTY?e!ky(>5dN80(_EuAP`6ZZDWZA 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-10,6 +16,9 @@ As $\quad f(x) \rightarrow \pm \infty,\quad {1 \over f(x)} \rightarrow 0^\pm$ (v +\includegraphics[width=0.25\textwidth]{./graphics/recip-parabola.png} +\includegraphics[width=0.25\textwidth]{./graphics/recip-sin-cos.png} + - reciprocal functions are always on the same side of $x=0$ - if $y=f(x)$ has a local max|min at $x=1$, then $y={1 \over f(x)}$ has a local max|min at $x=a$ - point of inflection at $P(1,1)$ @@ -107,6 +116,8 @@ $$x=f(t), \quad y=g(t)$$ $t$ is the parameter +To convert to cartesian, solve like simultaneous equations + ## Polar coordinates $$x = r\cos\theta, \quad y = r\sin\theta$$ @@ -132,25 +143,40 @@ $$r=a(n+ \cos\theta)$$ $$r=\cos(k\theta)$$ -where If $k$ is odd, half of the petals will overlap (hence there are $n$ petals) + If $k$ is even, petals will not overlap (hence $2n$ petals) +\includegraphics[width=0.5\textwidth]{./graphics/rose.png} + ### Solving polar graphs solve in terms of $r$ e.g. $x=4$ + $r\cos\theta = 4$ + $r={4 \over \cos\theta}$ + +--- + e.g. $y=x^2$ + $r\sin\theta = r^2 \cos^2 \theta$ + $\sin \theta = r \cos^2 \theta$ + $r = {\sin \theta \over \cos^2\theta} = \tan\theta \sec\theta$ -e.g. $r=6\cos \theta\quad$ *(multiple by $r$)* +--- + +e.g. $r=6\cos \theta\quad$ *(multiply by $r$)* + $r^2=6r\cos\theta$ + $x^2+y^2=6x$ + complete the square diff --git a/spec/graphing.pdf b/spec/graphing.pdf new file mode 100644 index 0000000000000000000000000000000000000000..07fb0398367b6af98d2bd5811e57862a879f6752 GIT binary patch literal 398169 zcma%iWk6MH6RwH^N(m?>-3Zd1g3{gH-Q6wH-QAnolz?!qCdt!Gw^RnT?(h`rjKm zAyW$nLpwq`Aq!mx!>@+=)&_=eczNI0JJ=cOTE20aAJPzy{ZNbAa-dAF$YC?E{qk5? zu1rBsovOet%0O}J?H2vIUA6u(Em@5DVZ-Uh8H*1#a=RU=X9;<*ix1q+wH*+vDK|^G=r&A>QtX-+3isa}xd~cNdf2z^lxq2l zyLG{oKCTxS-SxEZGM9%jtB|Tv?TpAwg*Eoub1}tBj>2zvT|OKb6J(6#`m}|qQ;M7> zN0-ShjwIBMJYHC{Hg-mOXI6;0vE5cZ1Mxt3;8}aeF-~29r+2g@5-}1dYNm*J8 z{b7=2?LE3_#w=A#ak6eDBdAOL=j}kv_rl3v!TtOEe&ZT$jA_L*v97Ts6Xtm?ve>0( zuinNxr+<5j>)!Msx6aF@cv()$ 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