A.1: Trigonometry
- Page ID
- 92433
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A.1.1 Trigonometry — Graphs
A.1.2 Trigonometry — Special Triangles
From the above pair of special triangles we have
\[\begin{align*} \sin \frac{\pi}{4} &= \frac{1}{\sqrt{2}} & \sin \frac{\pi}{6} &= \frac{1}{2} & \sin \frac{\pi}{3} &= \frac{\sqrt{3}}{2}\\ \cos \frac{\pi}{4} &= \frac{1}{\sqrt{2}} & \cos \frac{\pi}{6} &= \frac{\sqrt{3}}{2} & \cos \frac{\pi}{3} &= \frac{1}{2}\\ \tan \frac{\pi}{4} &= 1 & \tan \frac{\pi}{6} &= \frac{1}{\sqrt{3}} & \tan \frac{\pi}{3} &= \sqrt{3} \end{align*}\]
A.1.3 Trigonometry — Simple Identities
- Periodicity
\[\begin{align*} \sin(\theta+2\pi) &= \sin(\theta) & \cos(\theta+2\pi) &= \cos(\theta) \end{align*}\]
- Reflection
\[\begin{align*} \sin(-\theta)&=-\sin(\theta) & \cos(-\theta) &=\cos(\theta) \end{align*}\]
- Reflection around \(\pi/4\)
\[\begin{align*} \sin\left(\tfrac{\pi}{2}-\theta\right)&=\cos\theta & \cos\left(\tfrac{\pi}{2}-\theta\right)&=\sin\theta \end{align*}\]
- Reflection around \(\pi/2\)
\[\begin{align*} \sin\left(\pi-\theta\right)&=\sin\theta & \cos\left(\pi-\theta\right)&=-\cos\theta \end{align*}\]
- Rotation by \(\pi\)
\[\begin{align*} \sin\left(\theta+\pi\right)&=-\sin\theta & \cos\left(\theta+\pi\right)&=-\cos\theta \end{align*}\]
- Pythagoras
\[\begin{align*} \sin^2\theta + \cos^2 \theta &=1\\ \tan^2\theta + 1 &= \sec^2\theta\\ 1 + \cot^2 \theta &=\csc^2\theta \end{align*}\]
- \(\sin\) and \(\cos\) building blocks
\[\begin{gather*} \tan\theta=\frac{\sin\theta}{\cos\theta}\quad \csc\theta=\frac{1}{\sin\theta}\quad \sec\theta=\frac{1}{\cos\theta}\quad \cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{1}{\tan\theta} \end{gather*}\]
A.1.4 Trigonometry — Add and Subtract Angles
- Sine
\[\begin{align*} \sin(\alpha \pm \beta) &= \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta) \end{align*}\]
- Cosine
\[\begin{align*} \cos(\alpha \pm \beta) &= \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta) \end{align*}\]
- Tangent
\[\begin{align*} \tan(\alpha+\beta)&=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\\ \tan(\alpha-\beta)&=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta} \end{align*}\]
- Double angle
\[\begin{align*} \sin(2\theta) &= 2\sin(\theta)\cos(\theta)\\ \cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta)\\ &= 2\cos^2(\theta) - 1\\ &= 1 - 2\sin^2(\theta)\\ \tan(2\theta) &= \frac{2\tan(\theta)}{1-\tan^2\theta}\\ \cos^2\theta&=\frac{1+\cos(2\theta)}{2}\\ \sin^2\theta&=\frac{1-\cos(2\theta)}{2}\\ \tan^2\theta&=\frac{1-\cos(2\theta)}{1+\cos(2\theta)} \end{align*}\]
- Products to sums
\[\begin{align*} \sin(\alpha)\cos(\beta)&= \frac{\sin(\alpha+\beta) + \sin(\alpha-\beta)}{2}\\ \sin(\alpha)\sin(\beta)&= \frac{\cos(\alpha-\beta) - \cos(\alpha+\beta)}{2}\\ \cos(\alpha)\cos(\beta)&= \frac{\cos(\alpha-\beta) + \cos(\alpha+\beta)}{2} \end{align*}\]
- Sums to products
\[\begin{align*} \sin\alpha+\sin\beta &= 2 \sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\\ \sin\alpha-\sin\beta &= 2 \cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}\\ \cos\alpha+\cos\beta &= 2 \cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\\ \cos\alpha-\cos\beta &= -2 \sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \end{align*}\]
A.1.5 Inverse Trigonometric Functions
\[ \arcsin x \nonumber \]
Domain: \(-1 \leq x \leq 1\)
Range: \(-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}\)
\[ \arccos x \nonumber \]
Domain: \(-1 \leq x \leq 1\)
Range: \(0 \leq \arccos x \leq \pi\)
\(\arctan x\)
Domain: all real numbers
Range: \(-\frac{\pi}{2} \lt \arctan x \lt \frac{\pi}{2}\)
Since these functions are inverses of each other we have
\[\begin{align*} \arcsin(\sin \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\\ \arccos(\cos \theta) &= \theta & 0 \leq \theta \leq \pi\\ \arctan(\tan \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \end{align*}\]
and also
\[\begin{align*} \sin(\arcsin x) &= x & -1 \leq x \leq 1\\ \cos(\arccos x) &= x & -1 \leq x \leq 1\\ \tan(\arctan x) &= x & \text{any real } x \end{align*}\]
\[ \textrm{arccsc} x \nonumber \]
Domain: \(|x|\ge 1\)
Range: \(-\frac{\pi}{2} \leq \textrm{arccsc} x \leq \frac{\pi}{2}\)
\[ \textrm{arccsc} x \ne 0 \nonumber \]
\[ \textrm{arcsec} x \nonumber \]
Domain: \(|x|\ge 1\)
Range: \(0 \leq \textrm{arcsec} x \leq \pi\)
\[ \textrm{arcsec} x \ne \frac{\pi}{2} \nonumber \]
\[ \textrm{arccot} x \nonumber \]
Domain: all real numbers
Range: \(0 \lt \textrm{arccot} x \lt \pi\)
Again
\[\begin{align*} \textrm{arccsc}(\csc \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2},\ \theta\ne 0\\ \textrm{arcsec}(\sec \theta) &= \theta & 0 \leq \theta \leq \pi,\ \theta\ne \frac{\pi}{2}\\ \textrm{arccot}(\cot \theta) &= \theta & 0 \lt \theta \lt \pi \end{align*}\]
and
\[\begin{align*} \csc(\textrm{arccsc} x) &= x & |x|\ge 1\\ \sec(\textrm{arcsec} x) &= x & |x|\ge 1\\ \cot(\textrm{arccot} x) &= x & \text{any real } x \end{align*}\]