Ratkaise (2)3/sin^{220^{circ}}-1/cos^220^circ+64sin^220^circ

$$( 2 ) \frac { 3 } { \sin ^ { 2 } 20 ^ { \circ } } - \frac { 1 } { \cos ^ { 2 } 20 ^ { \circ } } + 64 \sin ^ { 2 } 20 ^ { \circ }$$

Laske

$$\frac{5814915970713618737501269924237505759233698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000}\approx 57,645896511$$

Ratkaisun vaiheet

Evaluate trigonometric functions in the problem

$${(2)} \frac{3}{0,3420201433256687 ^ {2}} - \frac{1}{0,9396926207859084 ^ {2}} + 64 \cdot 0,3420201433256687 ^ {2}$$

Laske 0,3420201433256687 potenssiin 2, jolloin ratkaisuksi tulee 0,11697777844051095979530830215969.

$$2\times \frac{3}{0,11697777844051095979530830215969}-\frac{1}{0,9396926207859084^{2}}+64\times 0,3420201433256687^{2}$$

Lavenna \frac{3}{0,11697777844051095979530830215969} kertomalla sekä osoittaja että nimittäjä luvulla 100000000000000000000000000000000.

$$2\times \frac{300000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0,9396926207859084^{2}}+64\times 0,3420201433256687^{2}$$

Ilmaise 2\times \frac{300000000000000000000000000000000}{11697777844051095979530830215969} säännöllisenä murtolukuna.

$$\frac{2\times 300000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0,9396926207859084^{2}}+64\times 0,3420201433256687^{2}$$

Kerro 2 ja 300000000000000000000000000000000, niin saadaan 600000000000000000000000000000000.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0,9396926207859084^{2}}+64\times 0,3420201433256687^{2}$$

Laske 0,9396926207859084 potenssiin 2, jolloin ratkaisuksi tulee 0,88302222155948904756966801319056.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0,88302222155948904756966801319056}+64\times 0,3420201433256687^{2}$$

Lavenna \frac{1}{0,88302222155948904756966801319056} kertomalla sekä osoittaja että nimittäjä luvulla 100000000000000000000000000000000.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{100000000000000000000000000000000}{88302222155948904756966801319056}+64\times 0,3420201433256687^{2}$$

Supista murtoluku \frac{100000000000000000000000000000000}{88302222155948904756966801319056} luvulla 16.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{6250000000000000000000000000000}{5518888884746806547310425082441}+64\times 0,3420201433256687^{2}$$

Lukujen 11697777844051095979530830215969 ja 5518888884746806547310425082441 pienin yhteinen jaettava on 64558736119771056212369977545857700093089334767336350659700329. Muunna \frac{600000000000000000000000000000000}{11697777844051095979530830215969} ja \frac{6250000000000000000000000000000}{5518888884746806547310425082441} murtoluvuiksi, joiden nimittäjä on 64558736119771056212369977545857700093089334767336350659700329.

$$\frac{3311333330848083928386255049464600000000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}-\frac{73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0,3420201433256687^{2}$$

Koska arvoilla \frac{3311333330848083928386255049464600000000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} ja \frac{73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} on sama nimittäjä, laske niiden erotus vähentämällä niiden osoittajat toisistaan.

$$\frac{3311333330848083928386255049464600000000000000000000000000000000-73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0,3420201433256687^{2}$$

Vähennä 73111111525319349872067688849806250000000000000000000000000000 luvusta 3311333330848083928386255049464600000000000000000000000000000000 saadaksesi tuloksen 3238222219322764578514187360614793750000000000000000000000000000.

$$\frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0,3420201433256687^{2}$$

Laske 0,3420201433256687 potenssiin 2, jolloin ratkaisuksi tulee 0,11697777844051095979530830215969.

$$\frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0,11697777844051095979530830215969$$

Kerro 64 ja 0,11697777844051095979530830215969, niin saadaan 7,48657782019270142689973133822016.

$$\frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+7,48657782019270142689973133822016$$

Muunna desimaaliluku 7,48657782019270142689973133822016 murtoluvuksi \frac{11697777844051095979530830215969}{10000000000}. Supista murtoluku \frac{11697777844051095979530830215969}{10000000000} luvulla 1.

$$\frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+\frac{11697777844051095979530830215969}{1562500000000000000000000000000}$$

Lukujen 64558736119771056212369977545857700093089334767336350659700329 ja 1562500000000000000000000000000 pienin yhteinen jaettava on 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000. Muunna \frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} ja \frac{11697777844051095979530830215969}{1562500000000000000000000000000} murtoluvuiksi, joiden nimittäjä on 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000.

$$\frac{5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000}+\frac{755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000}$$

Koska arvoilla \frac{5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000} ja \frac{755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000} on sama nimittäjä, laske ne yhteen laskemalla niiden osoittajat yhteen.

$$\frac{5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000+755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000}$$

Selvitä 5814915970713618737501269924237505759233698368644990171879009323293911055544430132267890353801 laskemalla yhteen 5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000 ja 755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801.

$$\frac{5814915970713618737501269924237505759233698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000}$$