100倍角の公式とは？角度を100倍すると…！
1. 100倍角の公式
まさか、100倍角の公式を手計算で求めようと思っている方はあまりいないと思いますが、実際にどうなるのかは気になりますよね！なので、プログラムで実際に求めてみたら、次のようになりました。
1.1. $\sin 100 x$
また、画像だけでなく数式を書くと、
$$\begin{align*}&\sin 100 x \\ & =- 633825300114114700748351602688 \sin^{99}{\left(x \right)} \cos{\left(x \right)} +\\& 15528719852795810168334614265856 \sin^{97}{\left(x \right)} \cos{\left(x \right)} – 184443162333207377917770316382208 \sin^{95}{\left(x \right)} \cos{\left(x \right)} +\\& 1415014982504761069420694953000960 \sin^{93}{\left(x \right)} \cos{\left(x \right)} – 7882075644733551894507464855388160 \sin^{91}{\left(x \right)} \cos{\left(x \right)} +\\& 33975894489667257903166387981910016 \sin^{89}{\left(x \right)} \cos{\left(x \right)} – 117951775551007962720566999412375552 \sin^{87}{\left(x \right)} \cos{\left(x \right)} +\\& 338907520995638086434440203380719616 \sin^{85}{\left(x \right)} \cos{\left(x \right)} – 821942832849475522127005112818728960 \sin^{83}{\left(x \right)} \cos{\left(x \right)} +\\& 1707613834057854213552013674555637760 \sin^{81}{\left(x \right)} \cos{\left(x \right)} – 3073704901304137584393624614200147968 \sin^{79}{\left(x \right)} \cos{\left(x \right)} +\\& 4836611236423926403226127393437515776 \sin^{77}{\left(x \right)} \cos{\left(x \right)} – 6700721817128981371136197326324891648 \sin^{75}{\left(x \right)} \cos{\left(x \right)} +\\& 8220381539581310037534459584682393600 \sin^{73}{\left(x \right)} \cos{\left(x \right)} – 8971413075589569260232790609861017600 \sin^{71}{\left(x \right)} \cos{\left(x \right)} +\\& 8742729997192188083011170457060638720 \sin^{69}{\left(x \right)} \cos{\left(x \right)} – 7630373725228003438520909930157834240 \sin^{67}{\left(x \right)} \cos{\left(x \right)} +\\& 5978297769836681645134561252862853120 \sin^{65}{\left(x \right)} \cos{\left(x \right)} – 4212350732134247229634108199849164800 \sin^{63}{\left(x \right)} \cos{\left(x \right)} +\\& 2672748856354186107691817191132364800 \sin^{61}{\left(x \right)} \cos{\left(x \right)} – 1528478252227550180336257956178821120 \sin^{59}{\left(x \right)} \cos{\left(x \right)} +\\& 788193577324092332295158940030730240 \sin^{57}{\left(x \right)} \cos{\left(x \right)} – 366537572671693287396000486098206720 \sin^{55}{\left(x \right)} \cos{\left(x \right)} +\\& 153672584815772030430000203798937600 \sin^{53}{\left(x \right)} \cos{\left(x \right)} – 58048470909027921582384945404313600 \sin^{51}{\left(x \right)} \cos{\left(x \right)} +\\& 19736480109069493338010881437466624 \sin^{49}{\left(x \right)} \cos{\left(x \right)} – 6031730927303982371491891000639488 \sin^{47}{\left(x \right)} \cos{\left(x \right)} +\\& 1654059140643228042512109125238784 \sin^{45}{\left(x \right)} \cos{\left(x \right)} – 406130592568649742581098222714880 \sin^{43}{\left(x \right)} \cos{\left(x \right)} +\\& 89056805509832617180847910420480 \sin^{41}{\left(x \right)} \cos{\left(x \right)} – 17387281075729225259117925367808 \sin^{39}{\left(x \right)} \cos{\left(x \right)} +\\& 3011681925459409985274984267776 \sin^{37}{\left(x \right)} \cos{\left(x \right)} – 460886985835470369989232427008 \sin^{35}{\left(x \right)} \cos{\left(x \right)} +\\& 62014418039824710570690478080 \sin^{33}{\left(x \right)} \cos{\left(x \right)} – 7295813887038201243610644480 \sin^{31}{\left(x \right)} \cos{\left(x \right)} +\\& 745616144499508478742626304 \sin^{29}{\left(x \right)} \cos{\left(x \right)} – 65694478009288290444771328 \sin^{27}{\left(x \right)} \cos{\left(x \right)} +\\& 4946109348189658933100544 \sin^{25}{\left(x \right)} \cos{\left(x \right)} – 314905094324468947353600 \sin^{23}{\left(x \right)} \cos{\left(x \right)} +\\& 16744638264836200857600 \sin^{21}{\left(x \right)} \cos{\left(x \right)} – 732577924086583787520 \sin^{19}{\left(x \right)} \cos{\left(x \right)} +\\& 25893101492105379840 \sin^{17}{\left(x \right)} \cos{\left(x \right)} – 722795936561233920 \sin^{15}{\left(x \right)} \cos{\left(x \right)} +\\& 15482165103820800 \sin^{13}{\left(x \right)} \cos{\left(x \right)} – 245050502860800 \sin^{11}{\left(x \right)} \cos{\left(x \right)} +\\& 2722783365120 \sin^{9}{\left(x \right)} \cos{\left(x \right)} – 19730314240 \sin^{7}{\left(x \right)} \cos{\left(x \right)} +\\& 83166720 \sin^{5}{\left(x \right)} \cos{\left(x \right)} – 166600 \sin^{3}{\left(x \right)} \cos{\left(x \right)} +\\& 100 \sin{\left(x \right)} \cos{\left(x \right)} \end{align*}$$
1.2. $\cos 100 x$
$$\begin{align*}&\cos 100 x \\ & =633825300114114700748351602688 \cos^{100}{\left(x \right)} – 15845632502852867518708790067200 \cos^{98}{\left(x \right)} +\\& 192128294097091018664344079564800 \cos^{96}{\left(x \right)} – 1505335087771022414277335056384000 \cos^{94}{\left(x \right)} +\\& 8567473526884295537508113973248000 \cos^{92}{\left(x \right)} – 37750993877408064336851542202122240 \cos^{90}{\left(x \right)} +\\& 134036108580690866727917044786790400 \cos^{88}{\left(x \right)} – 394078512785625681900511864396185600 \cos^{86}{\left(x \right)} +\\& 978503372439851812055958467641344000 \cos^{84}{\left(x \right)} – 2082455895192505138478065456775168000 \cos^{82}{\left(x \right)} +\\& 3842131126630171980492030767750184960 \cos^{80}{\left(x \right)} – 6200783636440931286187342812099379200 \cos^{78}{\left(x \right)} +\\& 8816739233064449172547628060953804800 \cos^{76}{\left(x \right)} – 11108623702136905456127648087408640000 \cos^{74}{\left(x \right)} +\\& 12460295938318846194767764735918080000 \cos^{72}{\left(x \right)} – 12489614281703125832873100652943769600 \cos^{70}{\left(x \right)} +\\& 11221137831217652115471926367879168000 \cos^{68}{\left(x \right)} – 9058026923994972189597820080095232000 \cos^{66}{\left(x \right)} +\\& 6581798018959761296303294062264320000 \cos^{64}{\left(x \right)} – 4310885252184171141438414824407040000 \cos^{62}{\left(x \right)} +\\& 2547463753712583633893763260298035200 \cos^{60}{\left(x \right)} – 1358954443662228159129584379363328000 \cos^{58}{\left(x \right)} +\\& 654531379770880870350000868032512000 \cos^{56}{\left(x \right)} – 284578860769948204500000377405440000 \cos^{54}{\left(x \right)} +\\& 111631674825053695350740279623680000 \cos^{52}{\left(x \right)} – 39472960218138986676021762874933248 \cos^{50}{\left(x \right)} +\\& 12566106098549963273941439584665600 \cos^{48}{\left(x \right)} – 3595780740528756614156758967910400 \cos^{46}{\left(x \right)} +\\& 923024074019658505866132324352000 \cos^{44}{\left(x \right)} – 212040013118649088525828358144000 \cos^{42}{\left(x \right)} +\\& 43468202689323063147794813419520 \cos^{40}{\left(x \right)} – 7925478751208973645460484915200 \cos^{38}{\left(x \right)} +\\& 1280241627320751027747867852800 \cos^{36}{\left(x \right)} – 182395347175955031090266112000 \cos^{34}{\left(x \right)} +\\& 22799418396994378886283264000 \cos^{32}{\left(x \right)} – 2485387148331694929142087680 \cos^{30}{\left(x \right)} +\\& 234623135747458180159897600 \cos^{28}{\left(x \right)} – 19023497493037149742694400 \cos^{26}{\left(x \right)} +\\& 1312104559685287280640000 \cos^{24}{\left(x \right)} – 76111992112891822080000 \cos^{22}{\left(x \right)} +\\& 3662889620432918937600 \cos^{20}{\left(x \right)} – 143850563845029888000 \cos^{18}{\left(x \right)} +\\& 4517474603507712000 \cos^{16}{\left(x \right)} – 110586893598720000 \cos^{14}{\left(x \right)} +\\& 2042087523840000 \cos^{12}{\left(x \right)} – 27227833651200 \cos^{10}{\left(x \right)} +\\& 246628928000 \cos^{8}{\left(x \right)} – 1386112000 \cos^{6}{\left(x \right)} +\\& 4165000 \cos^{4}{\left(x \right)} – 5000 \cos^{2}{\left(x \right)} +\\& 1 \end{align*}$$
