توابع مثلثاتی

${\displaystyle \sin A=\frac{BC}{AB}=\frac{a}{c}}$

${\displaystyle \cos A=\frac{AC}{AB}=\frac{b}{c}}$

${\displaystyle \tan A=\frac{a}{b}=\frac{a}{c}\times \frac{c}{b}=\frac{a}{c}/\frac{b}{c}=\frac{\sin A}{\cos A}}$

${\displaystyle \sin A=\cos B=\cos \left(\frac{\pi }{2}-A\right)}$

${\displaystyle \cos A=\sin B=\sin \left(\frac{\pi }{2}-A\right)}$

${\displaystyle \tan A=\cot B=\cot \left(\frac{\pi }{2}-A\right)}$

${\displaystyle \cot A=\tan B=\tan \left(\frac{\pi }{2}-A\right)}$

${\displaystyle \sec A=\csc B=\csc \left(\frac{\pi }{2}-A\right)}$

${\displaystyle \csc A=\sec B=\sec \left(\frac{\pi }{2}-A\right)}$

${\displaystyle \sin 0^{\circ }=\cos {90}^{\circ }=0}$

${\displaystyle \sin {90}^{\circ }=\cos 0^{\circ }=1}$

${\displaystyle \sin {45}^{\circ }=\cos {45}^{\circ }=\frac{\sqrt{2}}{2}}$

${\displaystyle \tan {45}^{\circ }=\cot {45}^{\circ }=1}$

${\displaystyle \sin {30}^{\circ }=\cos {60}^{\circ }=\frac{1}{2}}$

${\displaystyle \sin {60}^{\circ }=\cos {30}^{\circ }=\frac{\sqrt{3}}{2}}$

${\displaystyle {\int }_0^{\sin \theta }\frac{dx}{\sqrt{1-x^2}}=\theta }$

${\displaystyle f(x)=f(0)+\frac{f^{\prime }(0)}{1!}x+\frac{f^{″}(0)}{2!}{x}^2+\frac{f^{(3)}(0)}{3!}{x}^3+\cdots }$

${\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}}$

${\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{x}^{2n}}{(2n)!}}$

${\displaystyle a{y}^{″}+b{y}^{\prime }+cy=0}$

${\displaystyle y=A_1.e^{(\alpha +i\beta )x}+A_2.e^{(\alpha -i\beta )x}}$

${\displaystyle y=A_1.e^{\alpha x}\cos \beta x+A_2.e^{\alpha x}\sin \beta x}$

${\displaystyle \sin (2\pi +x)=\sin (x)}$

${\displaystyle \tan (\pi +x)=\tan (x)}$

${\displaystyle {\int }_0^{2\pi }\cos n\theta \sin m\theta d\theta =0.}$

${\displaystyle {\int }_0^{2\pi }\sin m\theta \sin n\theta d\theta =0(n\ne m).}$

${\displaystyle \mathcal{L}\{\sin (at)\}=\frac{a}{s^2+a^2}}$

${\displaystyle \mathcal{L}\{\cos (at)\}=\frac{s}{s^2+a^2}}$

${\displaystyle \mathcal{F}\{\sin (at)\}=\frac{{\displaystyle \delta \left(\omega -\frac{a}{2\pi }\right)-\delta \left(\omega +\frac{a}{2\pi }\right)}}{2i}}$

${\displaystyle \mathcal{F}\{\cos (at)\}=\frac{\delta \left(\omega -\frac{a}{2\pi }\right)+\delta \left(\omega +\frac{a}{2\pi }\right)}{2}}$

${\displaystyle {\cos }^{2}a+{\sin }^{2}a=1}$

${\displaystyle \cos (a\pm b)=\cos a.\cos b\mp \sin a.\sin b}$

${\displaystyle \sin (a\pm b)=\sin a.\cos b\pm \cos a.\sin b}$

${\displaystyle \cos 2a=1-2{\sin }^{2}a}$

${\displaystyle \sin 2a=2\sin a.\cos a}$

${\displaystyle \frac{2}{\pi }x\le \sin x\le x0\le x\le \frac{\pi }{2}.}$

${\displaystyle x\le \tan x0\le x\le \frac{\pi }{2}.}$

${\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R=\frac{abc}{2\mathrm{\Delta }}}$

${\displaystyle \mathrm{\Delta }=\frac{1}{2}ab\sin C}$

${\displaystyle c^2=a^2+b^2-2ab\cos C}$

${\displaystyle x^2{y}^{″}+x{y}^{\prime }+(x^2-a^2)y=0}$

${\displaystyle J_{1/2}=\sqrt{\frac{2}{\pi x}}\sin x}$

${\displaystyle J_{-1/2}=\sqrt{\frac{2}{\pi x}}\cos x}$

${\displaystyle (1-x^2)y^{″}-x{y}^{\prime }+n^{2}y=0}$

${\displaystyle T_n=\cos (n\arccos x)}$

${\displaystyle \mathbf{x}\cdot \mathbf{y}=\cos \left(\mathrm{\angle }(\mathbf{x},\mathbf{y})\right)\cdot |\mathbf{x}|\cdot |\mathbf{y}|}$

${\displaystyle \mathbf{x}\times \mathbf{y}=\sin \left(\mathrm{\angle }(\mathbf{x},\mathbf{y})\right)\cdot |\mathbf{x}|\cdot |\mathbf{y}|}$

${\displaystyle x=r\cos \theta ,y=r\sin \theta }$

${\displaystyle z=|z|(\cos \theta +i\sin \theta )}$

${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$

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

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

${\displaystyle \cos \theta =\mathrm{Re}(e^{i\theta })}$

${\displaystyle \sin \theta =\mathrm{Im}(e^{i\theta })}$

${\displaystyle e^{in\theta }=(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )}$

${\displaystyle \sin z=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1)!}{z}^{2n+1}=\frac{e^{iz}-e^{-iz}}{2i}=\frac{\sinh \left(iz\right)}{i}}$

${\displaystyle \cos z=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n)!}{z}^{2n}=\frac{e^{iz}+e^{-iz}}{2}=\cosh \left(iz\right)}$

${\displaystyle \sin (x+iy)=\sin x\cosh y+i\cos x\sinh y}$

${\displaystyle \cos (x+iy)=\cos x\cosh y-i\sin x\sinh y}$

${\displaystyle s=R\arccos (\sin {\alpha }_1\sin {\alpha }_2+\cos {\alpha }_1\cos {\alpha }_2\cos \varphi )}$

${\displaystyle \frac{\sin \theta }{\sin {\theta }^{\mathrm{\prime }}}=\frac{n^{\mathrm{\prime }}}{n}}$

${\displaystyle f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos nx+b_n\sin nx)}$

${\displaystyle F(\omega )=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-\mathrm{i}\omega t}dt}$

${\displaystyle f(t)=\sum _{k=1}^{\mathrm{\infty }}{c}_k{\phi }_k(t)}$

${\displaystyle f_{\text{square}}(t)=\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{\sin \left((2k-1)t\right)}{2k-1}}$

${\displaystyle x(t)=v_0.t\cos \theta }$

${\displaystyle y(t)=-\frac{1}{2}g{t}^2+v_0.t\sin \theta }$