From: Andrew Lorimer Date: Thu, 2 Aug 2018 09:05:46 +0000 (+1000) Subject: render final notes for graphing & circ fn's, add local graphics X-Git-Tag: yr11~77 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/30b2c80b12f81ebba914334f500e199b3c3aaa89?ds=sidebyside render final notes for graphing & circ fn's, add local graphics --- 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

+ $1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi} +$ +

Exact values

$\sin$ and $\cos$ graphs

+ $f(x)=a \sin(bx-c)+d +$ + $f(x)=a \cos(bx-c)+d +$ +

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)

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

Solve domain for $n\theta$
Find solutions for $n\theta$
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

$a=0$ produces straight line
+ $a\lt0$ 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

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

+ $\sin(\theta+{\pi\over 2})=\sin\theta +$ + $\sin(\theta+\pi)=-\sin\theta +$ + $\cos(\theta+{\pi \over 2})=-\cos\theta +$ + $\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta) +$ +

Pythagorean identity

+ $\cos^2\theta+\sin^2\theta=1 +$ +

Complementary relationships

+ $\sin({\pi \over 2} - \theta)=\cos\theta +$ + $\cos({\pi \over 2} - \theta)=\sin\theta +$ + $\sin\theta=-\cos(\theta+{\pi \over 2}) +$ + $\cos\theta=\sin(\theta+{\pi \over 2}) +$ +

$\tan$ 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 $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$
+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 0000000..84f3121 Binary files /dev/null and b/methods/circ-functions.pdf differ diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex new file mode 100644 index 0000000..7870df1 --- /dev/null +++ b/methods/circ-functions.tex @@ -0,0 +1,215 @@ +\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere +\PassOptionsToPackage{hyphens}{url} +% +\documentclass[]{article} +\usepackage{lmodern} +\usepackage{amssymb,amsmath} +\usepackage{ifxetex,ifluatex} +\usepackage{fixltx2e} % provides \textsubscript +\ifnum 0\ifxetex 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 0000000..4fde496 Binary files /dev/null and b/methods/exact-values-1.png differ diff --git a/methods/exact-values-2.png b/methods/exact-values-2.png new file mode 100644 index 0000000..6bcb11e Binary files /dev/null and b/methods/exact-values-2.png differ diff --git a/methods/graphics/exact-values-1.png b/methods/graphics/exact-values-1.png new file mode 100644 index 0000000..459ab1b Binary files /dev/null and b/methods/graphics/exact-values-1.png differ diff --git a/methods/graphics/exact-values-2.png b/methods/graphics/exact-values-2.png new file mode 100644 index 0000000..db84bab Binary files /dev/null and b/methods/graphics/exact-values-2.png differ diff --git a/physics/midyear.fdb_latexmk b/physics/midyear.fdb_latexmk index 48a4005..400d1eb 100644 --- a/physics/midyear.fdb_latexmk +++ b/physics/midyear.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1532049248 "midyear.tex" "midyear.pdf" "midyear" 1532049249 +["pdflatex"] 1532049248 "midyear.tex" "midyear.pdf" "midyear" 1532764905 "/mnt/andrew/graphics/ac-generator.png" 1529320281 28991 f74a831edcaed66632e709a6b2e2ebfd "" "/mnt/andrew/graphics/ac-motor.png" 1529313736 18626 d0f4943be26fdce44a4ce45ccb01633c "" "/mnt/andrew/graphics/banked-track.png" 1531025364 51228 cc5d8f3f8efb06a64aca4afabfd63eed "" @@ -71,8 +71,8 @@ "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1530607795 2586890 ddabbb88c498d59d5daa7919afb0a27f "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1530607774 7906415 c2bc2a1e85cbebe301ae391211242f7e "" "midyear.aux" 1532049249 223 c9c8cc73e6c5d1c4bae220d8eb29b53c "" - "midyear.tex" 1532049248 9329 b74d77b1da90c209352ffa9ba1392fc8 "" + "midyear.tex" 1532764905 9329 b74d77b1da90c209352ffa9ba1392fc8 "" (generated) - "midyear.aux" "midyear.pdf" + "midyear.aux" "midyear.log" diff --git a/spec/graphics/recip-parabola.png b/spec/graphics/recip-parabola.png new file mode 100644 index 0000000..b9e7bf7 Binary files /dev/null and b/spec/graphics/recip-parabola.png differ diff --git a/spec/graphics/recip-sin-cos.png b/spec/graphics/recip-sin-cos.png new file mode 100644 index 0000000..b3adbbd Binary files /dev/null and b/spec/graphics/recip-sin-cos.png differ diff --git a/spec/graphics/rose.png b/spec/graphics/rose.png new file mode 100644 index 0000000..1887b5e Binary files /dev/null and b/spec/graphics/rose.png differ diff --git a/spec/graphing.md b/spec/graphing.md index 6829ab1..0255539 100644 --- a/spec/graphing.md +++ b/spec/graphing.md @@ -1,3 +1,9 @@ +--- +geometry: margin=2cm +columns: 2 +graphics: yes +--- + # Graphing techniques ## Reciprocal continuous functions @@ -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 0000000..07fb039 Binary files /dev/null and b/spec/graphing.pdf differ