75807917c13a6af9ac95a0b4f0ff84c699451711
1\section{Circular functions}
2
3\subsection*{Radians and degrees}
4
5\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
6
7\subsection*{Exact values}
8
9
10 \begin{tikzpicture}[scale=0.75]
11 \draw [orange, thick] (0,0) -- (3,3) node [black, pos=0.5, above left] {\(\sqrt{2}\)};
12 \draw [orange, thick] (0,0) -- (3,0) node [black, below, pos=0.5] {\(1\)} node[black, above, pos=0.3] {\(\frac{\pi}{4}\)};
13 \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\frac{\pi}{4}\)};
14 \draw [black] (0,0) coordinate (A) (3,0) coordinate (B) (3,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C};
15 \end{tikzpicture}
16 \begin{tikzpicture}[scale=0.75]
17 \draw [orange, thick] (0,3) -- (5.19,0) node [black, pos=0.5, above right] {2};
18 \draw [orange, thick] (0,0) -- (5.19,0) node [black, below, pos=0.5] {\(\sqrt{3}\)} node[black, above, pos=0.7] {\(\frac{\pi}{6}\)};
19 \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\frac{\pi}{3}\)};
20 \draw [black] (5.19,0) coordinate (A) (0,0) coordinate (B) (0,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C};
21 \end{tikzpicture}
22
23 \subsection*{Compound angle formulas}
24
25
26 \begin{align*}
27 \cos(x \pm y) &= \cos x + \cos y \mp \sin x \sin y \\
28 \sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y \\
29 \tan(x \pm y) &= {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}
30 \end{align*}
31
32 \subsection*{Double angle formulas}
33
34 \begin{align*}
35 \cos 2x &= \cos^2 x - \sin^2 x \\
36 & = 1 - 2\sin^2 x \\
37 & = 2 \cos^2 x -1 \\
38 \sin 2x &= 2 \sin x \cos x \\
39 \tan 2x &= \dfrac{2 \tan x}{1 - \tan^2 x}
40 \end{align*}
41
42
43
44\subsection*{Symmetry}
45
46\begin{align*}
47 \sin(\theta+\frac{\pi}{2}) &= \sin\theta \\
48 \sin(\theta+\pi) &= -\sin\theta \\ \\
49 \cos(\theta+\frac{\pi}{2}) &= -\cos\theta \\
50 \cos(\theta+\pi) &= -\cos(\theta+\frac{3\pi}{2}) \\
51 &= \cos(-\theta)
52\end{align*}
53
54\subsection*{Complementary relationships}
55
56\begin{align*}
57 \sin \theta &= \cos(\frac{\pi}{2} - \theta) \\
58 &= -\cos(\theta+\frac{\pi}{2}) \\
59 \cos\theta &= \sin(\frac{\pi}{2} - \theta) \\
60 &= \sin(\theta+\frac{\pi}{2})
61\end{align*}
62
63\subsection*{Pythagorean identity}
64
65\[\cos^2\theta+\sin^2\theta=1\]
66
67 \subsection*{Inverse circular functions}
68
69 \begin{tikzpicture}
70 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
71 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
72 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
73 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
74 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
75 \addplot[mark=*, blue] coordinates {(1,0)};
76 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
77 \end{axis}
78 \end{tikzpicture}\\
79
80 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
81
82 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
83 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
84
85 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
86 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
87
88 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
89 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
90
91 \begin{tikzpicture}
92 \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
93 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
94 \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
95 \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
96 \end{axis}
97 \end{tikzpicture}
98
99\subsection*{\(\sin\) and \(\cos\) graphs}
100
101\[ f(x)=a\sin(bx-c)+d \]
102
103where:
104\begin{description}
105 \item Period \(=\frac{2\pi}{n}\)
106 \item dom \(= \mathbb{R}\)
107 \item ran \(= [-b+c, b+c]\);
108 \item \(\cos(x)\) starts at \((0,1)\), \(\sin(x)\) starts at \((0,0)\)
109 \item 0 amplitidue \(\implies\) straight line
110 \item \(a<0\) or \(b<0\) inverts phase (swap \(\sin\) and \(\cos\))
111 \item \(c=T={{2\pi}\over b} \implies\) no net phase shift
112\end{description}
113
114\subsection*{\(\tan\) graphs}
115
116\[y=a\tan(nx)\]
117
118\begin{description}
119 \item Period \(= \dfrac{\pi}{n}\)
120 \item Range is \(\mathbb{R}\)
121 \item Roots at \(x={\dfrac{k\pi}{n}}\) where \(k \in \mathbb{Z}\)
122 \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
123\end{description}
124
125\textbf{Asymptotes should always have equations}
126
127\subsection*{Solving trig equations}
128
129\begin{enumerate}
130\def\labelenumi{\arabic{enumi}.}
131\tightlist
132\item
133 Solve domain for \(n\theta\)
134\item
135 Find solutions for \(n\theta\)
136\item
137 Divide solutions by \(n\)
138\end{enumerate}
139
140\(\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])\)
141
142\(2\theta=\sin^{-1}{\sqrt{3} \over 2}\)
143
144\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
145
146\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)