User:Sagie/sandbox/Trigo

Trigonometric functions

${\begin{aligned}\tan \theta ={\frac {\sin \theta }{\cos \theta }}\\\sec \theta ={\frac {1}{\cos \theta }}\\\csc \theta ={\frac {1}{\sin \theta }}\\\end{aligned}}$	$\cot \theta ={\frac {1}{\tan \theta }}={\frac {\cos \theta }{\sin \theta }}$ $\cos ^{2}\theta +\sin ^{2}\theta =1$ $1+\tan ^{2}\theta =\sec ^{2}\theta$ $1+\cot ^{2}\theta =\csc ^{2}\theta$
$\operatorname {atan2} (y,x)={\begin{cases}\arctan(y/x)&\quad x>0\\\operatorname {sgn}(y)\ \pi /2&\quad x=0,y\neq 0\\{\text{undefined}}&\quad x=0,y=0\\\arctan(y/x)+\pi &\quad x<0,y\geq 0\\\arctan(y/x)-\pi &\quad x<0,y<0\end{cases}}$

Symmetry, shifts, and periodicity

Reflected with $0$	Reflected with $\pi /2$ (co-function identities)	Reflected with $\pi$	Shift by $\pi /2$	Shift by $\pi$ Period for tan and cot	Shift by $2\pi$ Period for sin, cos, csc and sec
${\begin{aligned}\sin(-\theta )&=-\sin \theta \\\cos(-\theta )&=+\cos \theta \\\tan(-\theta )&=-\tan \theta \\\csc(-\theta )&=-\csc \theta \\\sec(-\theta )&=+\sec \theta \\\cot(-\theta )&=-\cot \theta \\\end{aligned}}$	${\begin{aligned}\sin({\tfrac {\pi }{2}}-\theta )&=+\cos \theta \\\cos({\tfrac {\pi }{2}}-\theta )&=+\sin \theta \\\tan({\tfrac {\pi }{2}}-\theta )&=+\cot \theta \\\csc({\tfrac {\pi }{2}}-\theta )&=+\sec \theta \\\sec({\tfrac {\pi }{2}}-\theta )&=+\csc \theta \\\cot({\tfrac {\pi }{2}}-\theta )&=+\tan \theta \\\end{aligned}}$	${\begin{aligned}\sin(\pi -\theta )&=+\sin \theta \\\cos(\pi -\theta )&=-\cos \theta \\\tan(\pi -\theta )&=-\tan \theta \\\csc(\pi -\theta )&=+\csc \theta \\\sec(\pi -\theta )&=-\sec \theta \\\cot(\pi -\theta )&=-\cot \theta \\\end{aligned}}$	${\begin{aligned}\sin(\theta +{\tfrac {\pi }{2}})&=+\cos \theta \\\cos(\theta +{\tfrac {\pi }{2}})&=-\sin \theta \\\tan(\theta +{\tfrac {\pi }{2}})&=-\cot \theta \\\csc(\theta +{\tfrac {\pi }{2}})&=+\sec \theta \\\sec(\theta +{\tfrac {\pi }{2}})&=-\csc \theta \\\cot(\theta +{\tfrac {\pi }{2}})&=-\tan \theta \end{aligned}}$	${\begin{aligned}\sin(\theta +\pi )&=-\sin \theta \\\cos(\theta +\pi )&=-\cos \theta \\\tan(\theta +\pi )&=+\tan \theta \\\csc(\theta +\pi )&=-\csc \theta \\\sec(\theta +\pi )&=-\sec \theta \\\cot(\theta +\pi )&=+\cot \theta \\\end{aligned}}$	${\begin{aligned}\sin(\theta +2\pi )&=+\sin \theta \\\cos(\theta +2\pi )&=+\cos \theta \\\tan(\theta +2\pi )&=+\tan \theta \\\csc(\theta +2\pi )&=+\csc \theta \\\sec(\theta +2\pi )&=+\sec \theta \\\cot(\theta +2\pi )&=+\cot \theta \end{aligned}}$

Angle sum and difference identities

$\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

$\cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

$\tan(\alpha \pm \beta )={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}$

$\tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}=-\,{\frac {\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta }}$

$\arctan \alpha \pm \arctan \beta =\arctan \left({\frac {\alpha \pm \beta }{1\mp \alpha \beta }}\right)$

$\arcsin \alpha \pm \arcsin \beta =\arcsin \left(\alpha {\sqrt {1-\beta ^{2}}}\pm \beta {\sqrt {1-\alpha ^{2}}}\right)$

$\arccos \alpha \pm \arccos \beta =\arccos \left(\alpha \beta \mp {\sqrt {(1-\alpha ^{2})(1-\beta ^{2})}}\right)$

Multiple-angle formulae

Double-angle formulae
$\sin 2\theta =2\sin \theta \cos \theta ={\frac {2\tan \theta }{1+\tan ^{2}\theta }}$	${\begin{aligned}\cos 2\theta &=\cos ^{2}\theta -\sin ^{2}\theta =1-2\sin ^{2}\theta \\&=2\cos ^{2}\theta -1={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}$	$\tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}$	$\cot 2\theta ={\frac {\cot ^{2}\theta -1}{2\cot \theta }}$
Triple-angle formulae
${\begin{aligned}\sin 3\theta &=-\sin ^{3}\theta +3\cos ^{2}\theta \sin \theta \\&=-4\sin ^{3}\theta +3\sin \theta \end{aligned}}$	${\begin{aligned}\cos 3\theta &=\cos ^{3}\theta -3\sin ^{2}\theta \cos \theta \\&=4\cos ^{3}\theta -3\cos \theta \end{aligned}}$	$\tan 3\theta ={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}$	$\cot 3\theta ={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}$
Half-angle formulae
$\cos ^{2}{\frac {\theta }{2}}={\frac {1+\cos \theta }{2}}\;\Longleftrightarrow \;\cos {\frac {\theta }{2}}=\operatorname {sgn} \left(\pi +\theta +4\pi \left\lfloor {\frac {\pi -\theta }{4\pi }}\right\rfloor \right){\sqrt {\frac {1+\cos \theta }{2}}}$			$\tan {\frac {\eta +\theta }{2}}={\frac {\sin \eta +\sin \theta }{\cos \eta +\cos \theta }}$
$\sin ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{2}}\;\Longleftrightarrow \;\sin {\frac {\theta }{2}}=\operatorname {sgn} \left(2\pi -\theta +4\pi \left\lfloor {\frac {\theta }{4\pi }}\right\rfloor \right){\sqrt {\frac {1\!-\!\cos \theta }{2}}}$		${\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}={\frac {1-\tan(\theta /2)}{1+\tan(\theta /2)}}$
$\tan {\frac {\theta }{2}}=\csc \theta -\cot \theta =\pm \,{\sqrt {1-\cos \theta \over 1+\cos \theta }}={\frac {\sin \theta }{1+\cos \theta }}={\frac {1-\cos \theta }{\sin \theta }}$		$\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)=\sec \theta +\tan \theta$
$\cot {\tfrac {\theta }{2}}=\csc \theta +\cot \theta =\pm \,{\sqrt {1+\cos \theta \over 1-\cos \theta }}={\frac {\sin \theta }{1-\cos \theta }}={\frac {1+\cos \theta }{\sin \theta }}$		$\tan {\tfrac {\theta }{2}}={\frac {\tan \theta }{1+{\sqrt {1+\tan ^{2}\theta }}}}\quad {\text{for}}\ \theta \in \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$

Sine, cosine, and tangent of multiple angles

$\cos n\theta +j\sin n\theta =(\cos(\theta )+j\sin(\theta ))^{n}$

$\tan \,(n{+}1)\theta ={\frac {\tan n\theta +\tan \theta }{1-\tan n\theta \,\tan \theta }}$

$\cot \,(n{+}1)\theta ={\frac {\cot n\theta \,\cot \theta -1}{\cot n\theta +\cot \theta }}$

$\cos nx=2\cdot \cos x\cdot \cos((n-1)x)-\cos((n-2)x)$

$\sin nx=2\cdot \cos x\cdot \sin((n-1)x)-\sin((n-2)x)$

$\sin n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\sin \left({\frac {1}{2}}(n-k)\pi \right)$

$\cos n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\cos \left({\frac {1}{2}}(n-k)\pi \right)$

Power-reduction formula

Sine	Cosine	$\sin \cdot \cos$
$\sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}\!$	$\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}\!$	$\sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos 4\theta }{8}}\!$
$\sin ^{3}\theta ={\frac {3\sin \theta -\sin 3\theta }{4}}\!$	$\cos ^{3}\theta ={\frac {3\cos \theta +\cos 3\theta }{4}}\!$	$\sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin 2\theta -\sin 6\theta }{32}}\!$
$\sin ^{4}\theta ={\frac {3-4\cos 2\theta +\cos 4\theta }{8}}\!$	$\cos ^{4}\theta ={\frac {3+4\cos 2\theta +\cos 4\theta }{8}}\!$	$\sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos 4\theta +\cos 8\theta }{128}}\!$
$\sin ^{5}\theta ={\frac {10\sin \theta -5\sin 3\theta +\sin 5\theta }{16}}\!$	$\cos ^{5}\theta ={\frac {10\cos \theta +5\cos 3\theta +\cos 5\theta }{16}}\!$	$\sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin 2\theta -5\sin 6\theta +\sin 10\theta }{512}}\!$

	Cosine	Sine
${\text{if }}n{\text{ is odd}}$	$\cos ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}{\binom {n}{k}}\cos {((n-2k)\theta )}$	$\sin ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}(-1)^{({\frac {n-1}{2}}-k)}{\binom {n}{k}}\sin {((n-2k)\theta )}$
${\text{if }}n{\text{ is even}}$	$\cos ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}{\binom {n}{k}}\cos {((n-2k)\theta )}$	$\sin ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{({\frac {n}{2}}-k)}{\binom {n}{k}}\cos {((n-2k)\theta )}$

Product-to-sum and sum-to-product identities

Product-to-sum
$\cos \theta \cos \varphi ={{\cos(\theta -\varphi )+\cos(\theta +\varphi )} \over 2}\quad \ \vert \ \quad \sin \theta \sin \varphi ={{\cos(\theta -\varphi )-\cos(\theta +\varphi )} \over 2}\quad \ \vert \ \quad \sin \theta \cos \varphi ={{\sin(\theta +\varphi )+\sin(\theta -\varphi )} \over 2}$
$\cos \theta \sin \varphi ={{\sin(\theta +\varphi )-\sin(\theta -\varphi )} \over 2}\quad \ \vert \ \quad \tan \theta \tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}$
$\prod _{k=1}^{n}\cos \theta _{k}={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\ {\text{ where }}S=\{1,-1\}^{n}$
Sum-to-product
$\sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)\quad \ \vert \ \quad \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)$ $\cos \theta -\cos \varphi =-2\sin \left({\theta +\varphi \over 2}\right)\sin \left({\theta -\varphi \over 2}\right)$

Linear combinations

• $a\sin x+b\cos x=c\cdot \sin(x+\varphi )=c\cdot \cos(x+\varphi -\pi /2)\,$, where $c={\sqrt {a^{2}+b^{2}}}$ and $\varphi =\operatorname {atan2} \left(b,a\right)$.
• $a\sin x+b\sin(x+\alpha )=c\sin(x+\beta )\,{\text{, where }}c={\sqrt {a^{2}+b^{2}+2ab\cos \alpha }}\,{\text{ and }}\beta =\arctan \left({\frac {b\sin \alpha }{a+b\cos \alpha }}\right)+{\begin{cases}0&{\text{if }}a+b\cos \alpha \geq 0\\\pi &{\text{if }}a+b\cos \alpha <0\end{cases}}.$
• $\sum _{i}a_{i}\sin(x+\delta _{i})=a\sin(x+\delta )\,$, where $a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\delta _{i}-\delta _{j})$ and $\tan \delta ={\frac {\sum _{i}a_{i}\sin \delta _{i}}{\sum _{i}a_{i}\cos \delta _{i}}}$.

Other sums of trigonometric functions

• ${\text{Lagrange identities: }}\sum _{n=1}^{N}\sin n\theta ={\frac {1}{2}}\cot {\frac {\theta }{2}}-{\frac {\cos(N+{\frac {1}{2}})\theta }{2\sin {\frac {1}{2}}\theta }}{\text{ and }}\sum _{n=1}^{N}\cos n\theta =-{\frac {1}{2}}+{\frac {\sin(N+{\frac {1}{2}})\theta }{2\sin {\frac {1}{2}}\theta }}.$
• ${\text{Dirichlet kernel: }}1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)=\sin \left(\left(n+{\frac {1}{2}}\right)x\right){\Big /}\sin(x/2).$
• ${\text{If }}\alpha \neq 0{\text{, then}}\sin {\varphi }+\sin {(\varphi +\alpha )}+\sin {(\varphi +2\alpha )}+\cdots +\sin {(\varphi +n\alpha )}=\sin \left({\frac {(n+1)\alpha }{2}}\right)\cdot \sin \left(\varphi +{\frac {n\alpha }{2}}\right){\Big /}\sin {\frac {\alpha }{2}}.$
• ${\text{If }}\alpha \neq 0{\text{, then}}\cos {\varphi }+\cos {(\varphi +\alpha )}+\cos {(\varphi +2\alpha )}+\cdots +\cos {(\varphi +n\alpha )}=\sin \left({\frac {(n+1)\alpha }{2}}\right)\cdot \cos \left(\varphi +{\frac {n\alpha }{2}}\right){\Big /}\sin {\frac {\alpha }{2}}.$

Inverse trigonometric functions

${\begin{aligned}\arcsin(x)+\arccos(x)&=\pi /2\\\arctan(x)+\operatorname {arccot}(x)&=\pi /2\\\arctan(x)+\arctan(1/x)&=\left\{{\begin{matrix}\pi /2,&{\mbox{if }}x>0\\-\pi /2,&{\mbox{if }}x<0\end{matrix}}\right.\\\end{aligned}}$

$\quad {\begin{aligned}\sin[\arccos(x)]&={\sqrt {1-x^{2}}}\\\sin[\arctan(x)]&={\frac {x}{\sqrt {1+x^{2}}}}\\\cos[\arctan(x)]&={\frac {1}{\sqrt {1+x^{2}}}}\\\cos[\arcsin(x)]&={\sqrt {1-x^{2}}}\\\end{aligned}}$

$\quad {\begin{aligned}\tan[\arcsin(x)]&={\frac {x}{\sqrt {1-x^{2}}}}\\\tan[\arccos(x)]&={\frac {\sqrt {1-x^{2}}}{x}}\\\cot[\arcsin(x)]&={\frac {\sqrt {1-x^{2}}}{x}}\\\cot[\arccos(x)]&={\frac {x}{\sqrt {1-x^{2}}}}\\\end{aligned}}$

Complex exponentials

$e^{i\theta }=\operatorname {cis} \,\theta =\cos \theta +i\sin \theta \,$

$\sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,$

$\cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,$

$\sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}\,$

$\csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\,$

$e^{i\pi /2}=i\quad \ \vert \ \quad i^{2}=e^{i\pi }=-1\quad \ \vert \ \quad e^{2\pi i}=1$ $e^{-i\theta }=\cos(-\theta )+i\sin(-\theta )=\cos(\theta )-i\sin(\theta )$

$\arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,$

$\arccos x=i\,\ln \left(x-i\,{\sqrt {1-x^{2}}}\right)\,$

$\operatorname {arcsec} x=-i\ln \left({\tfrac {1}{x}}+{\sqrt {1-{\tfrac {i}{x^{2}}}}}\right)\,$

$\operatorname {arccsc} x=-i\ln \left({\tfrac {i}{x}}+{\sqrt {1-{\tfrac {1}{x^{2}}}}}\right)\,$

$\operatorname {arccis} \,x={\frac {\ln x}{i}}=-i\ln x=\operatorname {arg} \,x\,$

$\tan \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,$

$\cot \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,$

$\arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)\,$

$\operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)\,$

Infinite product formulae

$\sin x=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)$

$\sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)$

$\cos x=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)$

$\cosh x=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)$

${\frac {\sin x}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right)$

$|\sin x|={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\tan \left(2^{n}x\right)\right|}}$

Calculus

Limits

$\lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1$

$\lim _{x\rightarrow 0}{\frac {1-\cos x}{x}}=0$

Derivatives

${\mathrm {d} \over \mathrm {d} x}\sin x=\cos x$

${\mathrm {d} \over \mathrm {d} x}\cos x=-\sin x$

${\mathrm {d} \over \mathrm {d} x}\arcsin x={1 \over {\sqrt {1-x^{2}}}}$

${\mathrm {d} \over \mathrm {d} x}\arccos x={-1 \over {\sqrt {1-x^{2}}}}$

${\mathrm {d} \over \mathrm {d} x}\tan x=\sec ^{2}x$

${\mathrm {d} \over \mathrm {d} x}\cot x=-\csc ^{2}x$

${\mathrm {d} \over \mathrm {d} x}\arctan x={1 \over 1+x^{2}}$

${\mathrm {d} \over \mathrm {d} x}\operatorname {arccot} x={-1 \over 1+x^{2}}$

${\mathrm {d} \over \mathrm {d} x}\sec x=\tan x\sec x$

${\mathrm {d} \over \mathrm {d} x}\csc x=-\csc x\cot x$

${\mathrm {d} \over \mathrm {d} x}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}$

${\mathrm {d} \over \mathrm {d} x}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1}}}$

Integrals

$\int {\frac {\mathrm {d} u}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}\left({\frac {u}{a}}\right)+C$

$\int {\frac {\mathrm {d} u}{a^{2}+u^{2}}}={\frac {1}{a}}\tan ^{-1}\left({\frac {u}{a}}\right)+C$

$\int {\frac {\mathrm {d} u}{u{\sqrt {u^{2}-a^{2}}}}}={\frac {1}{a}}\sec ^{-1}\left|{\frac {u}{a}}\right|+C$

Tangent half-angle substitution

$t\triangleq \tan {\frac {x}{2}}.$

$\sin x={\frac {2t}{1+t^{2}}}$

$\cos x={\frac {1-t^{2}}{1+t^{2}}}$

$e^{ix}={\frac {1+it}{1-it}}$

$\mathrm {d} x={\frac {2\,\mathrm {d} t}{1+t^{2}}}$

Phasor arithmetic

• Definition: $A\cdot \cos(\omega t+\theta )={\frac {A}{2}}\cdot (e^{i(\omega t+\theta )}+e^{-i(\omega t+\theta )})=\operatorname {Re} \left\{A\cdot e^{i(\omega t+\theta )}\right\}=\operatorname {Re} \left\{Ae^{i\theta }\cdot e^{i\omega t}\right\}$
• $[{\text{Phasor}}]\cdot [{\text{complex const}}]:\ \operatorname {Re} \{(Ae^{i\theta }\cdot e^{i\omega t})\cdot Be^{i\phi }\}=\operatorname {Re} \{(ABe^{i(\theta +\phi )})\cdot e^{i\omega t}\}=AB\cos(\omega t+(\theta +\phi ))$
• Derivative: $\operatorname {Re} \left\{{\frac {d}{dt}}(Ae^{i\theta }\cdot e^{i\omega t})\right\}=\operatorname {Re} \{Ae^{i\theta }\cdot i\omega e^{i\omega t}\}=\operatorname {Re} \{\omega Ae^{i(\theta +\pi /2)}\cdot e^{i\omega t}\}=\omega A\cdot \cos(\omega t+\theta +\pi /2)$
• $A_{1}\cos(\omega t+\theta _{1})+A_{2}\cos(\omega t+\theta _{2})=\operatorname {Re} \{A_{1}e^{i\theta _{1}}e^{i\omega t}\}+\operatorname {Re} \{A_{2}e^{i\theta _{2}}e^{i\omega t}\}=\operatorname {Re} \{(A_{3}e^{i\theta _{3}})e^{i\omega t}\}=A_{3}\cos(\omega t+\theta _{3}),$ where: $A_{3}^{2}=(A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2})^{2}+(A_{1}\sin \theta _{1}+A_{2}\sin \theta _{2})^{2}=A_{1}^{2}+A_{2}^{2}+2A_{1}A_{2}\cos(\theta _{1}-\theta _{2})$ and $\theta _{3}=\arctan \left({\frac {A_{1}\sin \theta _{1}+A_{2}\sin \theta _{2}}{A_{1}\cos \theta _{1}+A_{2}\cos \theta _{2}}}\right)$.