Skip to main content\(\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section B.2 Trigonometric Formulae
-
Trigonometric Identities: in right angled triangle
\(\vartriangle ABC,\)
-
\begin{equation*}
h^{2}=p^{2}+b^{2}
\end{equation*}
-
\begin{equation*}
\sin\theta=\frac{p}{h}; \, \quad \cos\theta=\frac{b}{h}; \, \quad \tan\theta=\frac{p}{b}
\end{equation*}
-
\begin{equation*}
\sin\theta \csc\theta=1; \, \quad \cos\theta\sec\theta=1; \, \quad \tan\theta\cot\theta=1;
\end{equation*}
-
\begin{equation*}
\frac{\sin\theta}{\cos\theta}=\tan\theta
\end{equation*}
-
-
\begin{equation*}
\sin2\theta = 2\sin\theta\cos\theta;
\end{equation*}
-
\begin{equation*}
\cos2\theta = \cos^{2}\theta-\sin^{2}\theta = 2\cos^{2}\theta-1 = 1-2\sin^{2}\theta =2\cos^{2}\theta-1.
\end{equation*}
-
If \(2\theta = \alpha,\) then
-
\begin{equation*}
\sin\alpha=2\sin\frac{\alpha}{2}\cdot\cos\frac{\alpha}{2};
\end{equation*}
-
\begin{equation*}
\cos\alpha=1-2\sin^{2}\frac{\alpha}{2} = 2\cos^{2}\frac{\alpha}{2}-1.
\end{equation*}
-
\begin{equation*}
\sin(-\theta) = -\sin\theta; \, \quad \text{ and } \, \cos(-\theta) = \cos\theta;
\end{equation*}
-
\begin{equation*}
\sin(\theta \pm \frac{\pi}{2}) = \pm \cos\theta; \, \quad \text{ and } \, \cos(\theta \pm \frac{\pi}{2}) = \mp \sin\theta.
\end{equation*}
-
-
\begin{equation*}
\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta;
\end{equation*}
-
\begin{equation*}
or, \, \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta
\end{equation*}
-
-
\begin{equation*}
2\sin \alpha\cdot\sin\beta = \cos(\alpha-\beta) - \cos(\alpha+\beta);
\end{equation*}
-
\begin{equation*}
or, \, 2\cos \alpha\cdot\cos\beta = \cos(\alpha+\beta) + \cos(\alpha-\beta)
\end{equation*}
-
-
\begin{equation*}
2\sin \alpha\cdot\cos\beta = \sin(\alpha+\beta) + \sin(\alpha-\beta);
\end{equation*}
-
\begin{equation*}
or, \, \quad 2\cos \alpha\cdot\sin\beta = \sin(\alpha+\beta) - \sin(\alpha-\beta)
\end{equation*}
-
-
\begin{equation*}
\sin\alpha+\sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2};
\end{equation*}
-
\begin{equation*}
or, \, \cos\alpha+\cos\beta = 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
\end{equation*}
-
-
\begin{equation*}
\sin\alpha-\sin\beta = 2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2};
\end{equation*}
-
\begin{equation*}
or, \, \cos\alpha-\cos\beta = -2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
\end{equation*}
-
-
\begin{equation*}
\sin^2 x-\sin^2 y = \sin (x+y)\sin(x-y)
\end{equation*}
-
\begin{equation*}
\cos^2 x-\cos^2 y = -\sin (x+y)\sin (x-y)
\end{equation*}
-
Trigonometric Laws: in any
\(\vartriangle A' B' C'\) with sides
\(a, \,b,\) and
\(c\) and angles
\(\alpha,\,\beta,\) and
\(\gamma\)
-
Law of sines:
\begin{equation*}
\frac{\sin\alpha}{a}=\frac{\sin\beta}{b}=\frac{\sin\gamma}{c}
\end{equation*}
-
Law of cosines:
\begin{equation*}
c^{2}=a^{2}+b^{2}-2ab\cos\gamma
\end{equation*}
