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

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

Calcola

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

Procedura della soluzione

Calcola funzioni trigonometriche nel problema

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

Calcola 0,3420201433256687 alla potenza di 2 e ottieni 0,11697777844051095979530830215969.

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

Espandi \frac{3}{0,11697777844051095979530830215969} moltiplicando numeratore e denominatore per 100000000000000000000000000000000.

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

Esprimi 2\times \frac{300000000000000000000000000000000}{11697777844051095979530830215969} come singola frazione.

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

Moltiplica 2 e 300000000000000000000000000000000 per ottenere 600000000000000000000000000000000.

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

Calcola 0,9396926207859084 alla potenza di 2 e ottieni 0,88302222155948904756966801319056.

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

Espandi \frac{1}{0,88302222155948904756966801319056} moltiplicando numeratore e denominatore per 100000000000000000000000000000000.

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

Riduci la frazione \frac{100000000000000000000000000000000}{88302222155948904756966801319056} ai minimi termini estraendo e annullando 16.

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

Il minimo comune multiplo di 11697777844051095979530830215969 e 5518888884746806547310425082441 è 64558736119771056212369977545857700093089334767336350659700329. Converti \frac{600000000000000000000000000000000}{11697777844051095979530830215969} e \frac{6250000000000000000000000000000}{5518888884746806547310425082441} in frazioni con il denominatore 64558736119771056212369977545857700093089334767336350659700329.

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

Poiché \frac{3311333330848083928386255049464600000000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} e \frac{73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} hanno lo stesso denominatore, calcolane la sottrazione sottraendo i numeratori.

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

Sottrai 73111111525319349872067688849806250000000000000000000000000000 da 3311333330848083928386255049464600000000000000000000000000000000 per ottenere 3238222219322764578514187360614793750000000000000000000000000000.

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

Calcola 0,3420201433256687 alla potenza di 2 e ottieni 0,11697777844051095979530830215969.

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

Moltiplica 64 e 0,11697777844051095979530830215969 per ottenere 7,48657782019270142689973133822016.

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

Converti il numero decimale 7,48657782019270142689973133822016 nella frazione \frac{11697777844051095979530830215969}{10000000000}. Riduci la frazione \frac{11697777844051095979530830215969}{10000000000} ai minimi termini estraendo e annullando 1.

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

Il minimo comune multiplo di 64558736119771056212369977545857700093089334767336350659700329 e 1562500000000000000000000000000 è 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000. Converti \frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} e \frac{11697777844051095979530830215969}{1562500000000000000000000000000} in frazioni con il denominatore 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000.

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

Poiché \frac{5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000} e \frac{755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000} hanno lo stesso denominatore, calcolane l'addizione sommando i numeratori.

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

E 5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000 e 755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801 per ottenere 5814915970713618737501269924237505759233698368644990171879009323293911055544430132267890353801.

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