Trigonometric Functions

f$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ F$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$ = L$\left\{\vphantom{ f}\right.$f$\left.\vphantom{ f}\right\}$$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$
sin kt ${\dfrac{{k}}{{s^{2}+k^{2}}}}$


cos kt ${\dfrac{{s}}{{s^{2}+k^{2}}}}$


sin2kt ${\dfrac{{2k^{2}}}{{s\left( s^{2}+4k^{2}\right) }}}$


cos2kt ${\dfrac{{s^{2}+2k^{2}}}{{s\left( s^{2}+4k^{2}\right) }}}$


t sin kt ${\dfrac{{2ks}}{{\left( s^{2}+k^{2}\right) ^{2}}}}$


t cos kt ${\dfrac{{s^{2}-k^{2}}}{{\left( s^{2}+k^{2}\right) ^{2}}}}$


${\dfrac{{2\left( 1-\cos kt\right) }}{{t}}}$     ln${\dfrac{{s^{2}+k^{2}}}{{s^{2}% }}}$


${\dfrac{{\sin at}}{{t}}}$ arctan$\left(\vphantom{ \dfrac{a}{s}}\right.$${\dfrac{{a}}{{s}}}$$\left.\vphantom{ \dfrac{a}{s}}\right)$
                
f$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ F$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$ = L$\left\{\vphantom{ f}\right.$f$\left.\vphantom{ f}\right\}$$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$
sin kt + kt cos kt ${\dfrac{{2ks^{2}}}{{\left( s^{2}+k^{2}\right) ^{2}}}}$


sin kt - kt cos kt ${\dfrac{{2k^{3}}}{{\left( s^{2}-k^{2}\right) ^{2}}}}$


1 - cos kt ${\dfrac{{k^{2}}}{{s\left( s^{2}+k^{2}\right) }}}$


kt - sin kt ${\dfrac{{k^{3}}}{{s^{2}\left( s^{2}+k^{2}\right) }}}$


${\dfrac{{a\sin bt-b\sin at}}{{ab\left( a^{2}-b^{2}\right) }}}$     ${\dfrac{{1}}{{% \left( s^{2}+a^{2}\right) \left( s^{2}+b^{2}\right) }}}$


${\dfrac{{\cos bt-\cos at}}{{a^{2}-b^{2}}}}$ ${\dfrac{{s}}{{\left( s^{2}+a^{2}\right) \left( s^{2}+b^{2}\right) }}}$


${\dfrac{{\sin at\cos bt}}{{t}}}$ ${\dfrac{{1}}{{2}}}$arctan${\dfrac{{a+b}}{{s}}}$ + ${\dfrac{{1}}{{2}}}$arctan${\dfrac{{a-b}}{{s}}}$