66.2: Trigonometry
- Page ID
- 91932
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
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 | \(\infty\) |