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

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

Evaluate

$$\frac{5814915970713618737501269924237505759233698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000}\approx 57.645896511$$

Solution Steps

Evaluate trigonometric functions in the problem

$${(2)} \frac{3}{0.3420201433256687 ^ {2}} - \frac{1}{0.9396926207859084 ^ {2}} + 64 \cdot 0.3420201433256687 ^ {2}$$

Calculate 0.3420201433256687 to the power of 2 and get 0.11697777844051095979530830215969.

$$2\times \frac{3}{0.11697777844051095979530830215969}-\frac{1}{0.9396926207859084^{2}}+64\times 0.3420201433256687^{2}$$

Expand \frac{3}{0.11697777844051095979530830215969} by multiplying both numerator and the denominator by 100000000000000000000000000000000.

$$2\times \frac{300000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0.9396926207859084^{2}}+64\times 0.3420201433256687^{2}$$

Express 2\times \frac{300000000000000000000000000000000}{11697777844051095979530830215969} as a single fraction.

$$\frac{2\times 300000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0.9396926207859084^{2}}+64\times 0.3420201433256687^{2}$$

Multiply 2 and 300000000000000000000000000000000 to get 600000000000000000000000000000000.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0.9396926207859084^{2}}+64\times 0.3420201433256687^{2}$$

Calculate 0.9396926207859084 to the power of 2 and get 0.88302222155948904756966801319056.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{1}{0.88302222155948904756966801319056}+64\times 0.3420201433256687^{2}$$

Expand \frac{1}{0.88302222155948904756966801319056} by multiplying both numerator and the denominator by 100000000000000000000000000000000.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{100000000000000000000000000000000}{88302222155948904756966801319056}+64\times 0.3420201433256687^{2}$$

Reduce the fraction \frac{100000000000000000000000000000000}{88302222155948904756966801319056} to lowest terms by extracting and canceling out 16.

$$\frac{600000000000000000000000000000000}{11697777844051095979530830215969}-\frac{6250000000000000000000000000000}{5518888884746806547310425082441}+64\times 0.3420201433256687^{2}$$

Least common multiple of 11697777844051095979530830215969 and 5518888884746806547310425082441 is 64558736119771056212369977545857700093089334767336350659700329. Convert \frac{600000000000000000000000000000000}{11697777844051095979530830215969} and \frac{6250000000000000000000000000000}{5518888884746806547310425082441} to fractions with denominator 64558736119771056212369977545857700093089334767336350659700329.

$$\frac{3311333330848083928386255049464600000000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}-\frac{73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0.3420201433256687^{2}$$

Since \frac{3311333330848083928386255049464600000000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} and \frac{73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3311333330848083928386255049464600000000000000000000000000000000-73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0.3420201433256687^{2}$$

Subtract 73111111525319349872067688849806250000000000000000000000000000 from 3311333330848083928386255049464600000000000000000000000000000000 to get 3238222219322764578514187360614793750000000000000000000000000000.

$$\frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0.3420201433256687^{2}$$

Calculate 0.3420201433256687 to the power of 2 and get 0.11697777844051095979530830215969.

$$\frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+64\times 0.11697777844051095979530830215969$$

Multiply 64 and 0.11697777844051095979530830215969 to get 7.48657782019270142689973133822016.

$$\frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}+7.48657782019270142689973133822016$$

Convert decimal number 7.48657782019270142689973133822016 to fraction \frac{11697777844051095979530830215969}{10000000000}. Reduce the fraction \frac{11697777844051095979530830215969}{10000000000} to lowest terms by extracting and canceling out 1.

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

Least common multiple of 64558736119771056212369977545857700093089334767336350659700329 and 1562500000000000000000000000000 is 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000. Convert \frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} and \frac{11697777844051095979530830215969}{1562500000000000000000000000000} to fractions with denominator 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000.

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

Since \frac{5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000} and \frac{755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000} have the same denominator, add them by adding their numerators.

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

Add 5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000 and 755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801 to get 5814915970713618737501269924237505759233698368644990171879009323293911055544430132267890353801.

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