Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

66.2: Trigonometry

Last updated

Aug 8, 2024
Save as PDF
- 66.1: Greek Alphabet
- 66.3: Hyperbolic Trigonometry

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Basic Formulæ

$\begin{aligned} & \sin ^{2} \theta+\cos ^{2} \theta \equiv 1 \\ & \sec ^{2} \theta \equiv 1+\tan ^{2} \theta \\ & \csc ^{2} \theta \equiv 1+\cot ^{2} \theta \end{aligned}$

Angle Addition Formulæ

$\begin{aligned} & \sin (\alpha \pm \beta) \equiv \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ & \cos (\alpha \pm \beta) \equiv \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ & \tan (\alpha \pm \beta) \equiv \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} \end{aligned}$

Double-Angle Formulæ

$\begin{aligned} & \sin 2 \theta \equiv 2 \sin \theta \cos \theta \equiv \frac{2 \tan \theta}{1+\tan ^{2} \theta} \\ & \cos 2 \theta \equiv \cos ^{2} \theta-\sin ^{2} \theta \equiv 1-2 \sin ^{2} \theta \equiv 2 \cos ^{2} \theta-1 \equiv \frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta} \\ & \tan 2 \theta \equiv \frac{2 \tan \theta}{1-\tan ^{2} \theta} \end{aligned}$

Triple-Angle Formulæ

$\begin{aligned} \sin 3 \theta & \equiv 3 \sin \theta-4 \sin ^{3} \theta \\ \cos 3 \theta & \equiv 4 \cos ^{3} \theta-3 \cos \theta \\ \tan 3 \theta & \equiv \frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta} \\ \cot 3 \theta & \equiv \frac{\cot ^{3} \theta-3 \cot \theta}{3 \cot ^{2} \theta-1} \end{aligned}$

Quadruple-Angle Formulæ

$\begin{aligned} \sin 4 \theta & \equiv 4 \cos ^{3} \theta \sin \theta-4 \cos \theta \sin ^{3} \theta \\ \cos 4 \theta & \equiv \cos ^{4} \theta-6 \cos ^{2} \theta \sin ^{2} \theta+\sin ^{4} \theta \\ \tan 4 \theta & \equiv \frac{4 \tan \theta-4 \tan ^{3} \theta}{1-6 \tan ^{2} \theta+\tan ^{4} \theta} \\ \cot 4 \theta & \equiv \frac{\cot ^{4} \theta-6 \cot ^{2} \theta+1}{4 \cot ^{3} \theta-4 \cot \theta} \end{aligned}$

Half-Angle Formulæ

$\begin{aligned} \sin \frac{\theta}{2} & \equiv \pm \sqrt{\frac{1-\cos \theta}{2}} \\ \cos \frac{\theta}{2} & \equiv \pm \sqrt{\frac{1+\cos \theta}{2}} \\ \tan \frac{\theta}{2} & \equiv \frac{\sin \theta}{1+\cos \theta} \equiv \frac{1-\cos \theta}{\sin \theta} \end{aligned}$

Products of Sines and Cosines

$\begin{aligned} \sin \alpha \cos \beta & \equiv \frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)] \\ \cos \alpha \sin \beta & \equiv \frac{1}{2}[\sin (\alpha+\beta)-\sin (\alpha-\beta)] \\ \cos \alpha \cos \beta & \equiv \frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)] \\ \sin \alpha \sin \beta & \equiv-\frac{1}{2}[\cos (\alpha+\beta)-\cos (\alpha-\beta)] \end{aligned}$

Sums and Differences of Sines and Cosines

$\begin{aligned} & \sin \alpha+\sin \beta \equiv 2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \\ & \sin \alpha-\sin \beta \equiv 2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \\ & \cos \alpha+\cos \beta \equiv 2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \\ & \cos \alpha-\cos \beta \equiv-2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \end{aligned}$

Power Reduction Formulæ

$\begin{aligned} \sin ^{2} \theta & \equiv \frac{1}{2}(1-\cos 2 \theta) \\ \cos ^{2} \theta & \equiv \frac{1}{2}(1+\cos 2 \theta) \\ \tan ^{2} \theta & \equiv \frac{1-\cos 2 \theta}{1+\cos 2 \theta} \end{aligned}$

Other Formulæ

$\tan \theta \equiv \cot \theta-2 \cot 2 \theta$

Trig Cheat Sheet Paul Dawkins

https://tutorial.math.lamar.edu/pdf/trig_cheat_sheet.pdf

Exact values of trigonometric functions at $3^{\circ}$ intervals. (Ref. [6])

\theta	\sin \theta	\cos \theta	\tan \theta
\(0^{\circ}=0 \pi	0	1	0
$3^{\circ}=\frac{\pi}{60}$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}-1)-2(\sqrt{3}-1) \sqrt{5+\sqrt{5}}]$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5+\sqrt{5}}+(\sqrt{6}-\sqrt{2})(\sqrt{5}-1)]$	$\frac{1}{4}(\sqrt{5}-\sqrt{3})(\sqrt{3}-1)(\sqrt{10+2 \sqrt{5}}-\sqrt{5}-1)$
$6^{\circ}=\frac{\pi}{30}$	$\frac{1}{8}(\sqrt{30-6 \sqrt{5}}-\sqrt{5}-1)$	$\frac{1}{8}(\sqrt{15}+\sqrt{3}+\sqrt{10-2 \sqrt{5}})$	$\frac{1}{2}(\sqrt{10-2 \sqrt{5}}-\sqrt{15}+\sqrt{3})$
$9^{\circ}=\frac{\pi}{20}$	$\frac{1}{8}(\sqrt{10}+\sqrt{2}-2 \sqrt{5-\sqrt{5}})$	$\frac{1}{8}(\sqrt{10}+\sqrt{2}+2 \sqrt{5-\sqrt{5}})$	$\sqrt{5}+1-\sqrt{5+2 \sqrt{5}}$
$12^{\circ}=\frac{\pi}{15}$	$\frac{1}{8}(\sqrt{10+2 \sqrt{5}}-\sqrt{15}+\sqrt{3})$	$\frac{1}{8}(\sqrt{30+6 \sqrt{5}}+\sqrt{5}-1)$	$\frac{1}{2}(3 \sqrt{3}-\sqrt{15}-\sqrt{50-22 \sqrt{5}})$
$15^{\circ}=\frac{\pi}{12}$	$\frac{1}{4}(\sqrt{6}-\sqrt{2})$	$\frac{1}{4}(\sqrt{6}+\sqrt{2})$	$2-\sqrt{3}$
$18^{\circ}=\frac{\pi}{10}$	$\frac{1}{4}(\sqrt{5}-1)$	$\frac{1}{4} \sqrt{10+2 \sqrt{5}}$	$\frac{1}{5} \sqrt{25-10 \sqrt{5}}$
$21^{\circ}=\frac{7 \pi}{60}$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5-\sqrt{5}}-(\sqrt{6}-\sqrt{2})(\sqrt{5}+1)]$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}+1)+2(\sqrt{3}-1) \sqrt{5-\sqrt{5}}]$	$\frac{1}{4}(\sqrt{5}-\sqrt{3})(\sqrt{3}+1)(\sqrt{10-2 \sqrt{5}}-\sqrt{5}+1)$
$24^{\circ}=\frac{2 \pi}{15}$ \)	$\frac{1}{8}(\sqrt{15}+\sqrt{3}-\sqrt{10-2 \sqrt{5}})$	$\frac{1}{8}(\sqrt{30-6 \sqrt{5}}+\sqrt{5}+1)$	$\frac{1}{2}(\sqrt{50+22 \sqrt{5}}-3 \sqrt{3}-\sqrt{15})$
$27^{\circ}=\frac{3 \pi}{20}$	$\frac{1}{8}(2 \sqrt{5+\sqrt{5}}-\sqrt{10}+\sqrt{2})$	$\frac{1}{8}(2 \sqrt{5+\sqrt{5}}+\sqrt{10}-\sqrt{2})$	$\sqrt{5}-1-\sqrt{5-2 \sqrt{5}}$
$30^{\circ}=\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{1}{2} \sqrt{3}$	$\frac{1}{3} \sqrt{3}$
$33^{\circ}=\frac{11 \pi}{60}$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}-1)+2(\sqrt{3}-1) \sqrt{5+\sqrt{5}}]$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5+\sqrt{5}}-(\sqrt{6}-\sqrt{2})(\sqrt{5}-1)]$	$\frac{1}{4}(\sqrt{5}-\sqrt{3})(\sqrt{3}-1)(\sqrt{10+2 \sqrt{5}}+\sqrt{5}+1)$
$36^{\circ}=\frac{\pi}{5}$	$\frac{1}{4} \sqrt{10-2 \sqrt{5}}$	$\frac{1}{4}(\sqrt{5}+1)$	$\sqrt{5-2 \sqrt{5}}$
$39^{\circ}=\frac{13 \pi}{60}$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}+1)-2(\sqrt{3}-1) \sqrt{5-\sqrt{5}}]$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5-\sqrt{5}}+(\sqrt{6}-\sqrt{2})(\sqrt{5}+1)]$	$\frac{1}{4}(\sqrt{5}+\sqrt{3})(\sqrt{3}-1)(\sqrt{10-2 \sqrt{5}}-\sqrt{5}+1)$
$42^{\circ}=\frac{7 \pi}{30}$	$\frac{1}{8}(\sqrt{30+6 \sqrt{5}}-\sqrt{5}+1)$	$\frac{1}{8}(\sqrt{10+2 \sqrt{5}}+\sqrt{15}-\sqrt{3})$	$\frac{1}{2}(\sqrt{15}+\sqrt{3}-\sqrt{10+2 \sqrt{5}})$
$45^{\circ}=\frac{\pi}{4}$	$\frac{1}{2} \sqrt{2}$	$\frac{1}{2} \sqrt{2}$	1
$48^{\circ}=\frac{4 \pi}{15}$	$\frac{1}{8}(\sqrt{10+2 \sqrt{5}}+\sqrt{15}-\sqrt{3})$	$\frac{1}{8}(\sqrt{30+6 \sqrt{5}}-\sqrt{5}+1)$	$\frac{1}{2}(3 \sqrt{3}-\sqrt{15}+\sqrt{50-22 \sqrt{5}})$
$51^{\circ}=\frac{17 \pi}{60}$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5-\sqrt{5}}+(\sqrt{6}-\sqrt{2})(\sqrt{5}+1)]$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}+1)-2(\sqrt{3}-1) \sqrt{5-\sqrt{5}}]$	$\frac{1}{4}(\sqrt{5}-\sqrt{3})(\sqrt{3}+1)(\sqrt{10-2 \sqrt{5}}+\sqrt{5}-1)$
$54^{\circ}=\frac{3 \pi}{10}$	$\frac{1}{4}(\sqrt{5}+1)$	$\frac{1}{4} \sqrt{10-2 \sqrt{5}}$	$\frac{1}{5} \sqrt{25+10 \sqrt{5}}$
$57^{\circ}=\frac{19 \pi}{60}$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5+\sqrt{5}}-(\sqrt{6}-\sqrt{2})(\sqrt{5}-1)]$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}-1)+2(\sqrt{3}-1) \sqrt{5+\sqrt{5}}]$	$\frac{1}{4}(\sqrt{5}+\sqrt{3})(\sqrt{3}+1)(\sqrt{10+2 \sqrt{5}}-\sqrt{5}-1)$
$60^{\circ}=\frac{\pi}{3}$	$\frac{1}{2} \sqrt{3}$	$\frac{1}{2}$	$\sqrt{3}$
$63^{\circ}=\frac{7 \pi}{20}$	$\frac{1}{8}(2 \sqrt{5+\sqrt{5}}+\sqrt{10}-\sqrt{2})$	$\frac{1}{8}(2 \sqrt{5+\sqrt{5}}-\sqrt{10}+\sqrt{2})$	$\sqrt{5}-1+\sqrt{5-2 \sqrt{5}}$
$66^{\circ}=\frac{11 \pi}{30}$	$\frac{1}{8}(\sqrt{30-6 \sqrt{5}}+\sqrt{5}+1)$	$\frac{1}{8}(\sqrt{15}+\sqrt{3}-\sqrt{10-2 \sqrt{5}})$	$\frac{1}{2}(\sqrt{10-2 \sqrt{5}}+\sqrt{15}-\sqrt{3})$
$69^{\circ}=\frac{23}{60} \pi$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}+1)+2(\sqrt{3}-1) \sqrt{5-\sqrt{5}}]$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5-\sqrt{5}}-(\sqrt{6}-\sqrt{2})(\sqrt{5}+1)]$	$\frac{1}{4}(\sqrt{5}+\sqrt{3})(\sqrt{3}-1)(\sqrt{10-2 \sqrt{5}}+\sqrt{5}-1)$
$75^{\circ}=\frac{5 \pi}{12}$	$\frac{1}{4}(\sqrt{6}+\sqrt{2})$	$\frac{1}{4}(\sqrt{6}-\sqrt{2})$	$2+\sqrt{3}$
$78^{\circ}=\frac{13 \pi}{30}$	$\frac{1}{8}(\sqrt{30+6 \sqrt{5}}+\sqrt{5}-1)$	$\frac{1}{8}(\sqrt{10+2 \sqrt{5}}-\sqrt{15}+\sqrt{3})$	$\frac{1}{2}(\sqrt{15}+\sqrt{3}+\sqrt{10+2 \sqrt{5}})$
$81^{\circ}=\frac{19 \pi}{20}$	$\frac{1}{8}(\sqrt{10}+\sqrt{2}+2 \sqrt{5-\sqrt{5}})$	$\frac{1}{8}(\sqrt{10}+\sqrt{2}-2 \sqrt{5-\sqrt{5}})$	$\sqrt{5}+1+\sqrt{5+2 \sqrt{5}}$
$84^{\circ}=\frac{7 \pi}{15}$	$\frac{1}{8}(\sqrt{15}+\sqrt{3}+\sqrt{10-2 \sqrt{5}})$	$\frac{1}{8}(\sqrt{30-6 \sqrt{5}}-\sqrt{5}-1)$	$\frac{1}{2}(\sqrt{50+22 \sqrt{5}}+3 \sqrt{3}+\sqrt{15})$
$87^{\circ}=\frac{29 \pi}{60}$	$\frac{1}{16}[2(\sqrt{3}+1) \sqrt{5+\sqrt{5}}+(\sqrt{6}-\sqrt{2})(\sqrt{5}-1)]$	$\frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}-1)-2(\sqrt{3}-1)$ \sqrt{5+\sqrt{5}}]\)	$\frac{1}{4}(\sqrt{5}+\sqrt{3})(\sqrt{3}+1)(\sqrt{10+2 \sqrt{5}}+\sqrt{5}+1)$
$90^{\circ}=\frac{\pi}{2}$	1	0	$\infty$