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

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

계산

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

해답 단계

Evaluate trigonometric functions in the problem

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

0.3420201433256687의 2제곱을 계산하여 0.11697777844051095979530830215969을(를) 구합니다.

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

분자와 분모 모두에 100000000000000000000000000000000을(를) 곱하여 \frac{3}{0.11697777844051095979530830215969}을(를) 확장합니다.

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

2\times \frac{300000000000000000000000000000000}{11697777844051095979530830215969}을(를) 단일 분수로 표현합니다.

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

2과(와) 300000000000000000000000000000000을(를) 곱하여 600000000000000000000000000000000(을)를 구합니다.

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

0.9396926207859084의 2제곱을 계산하여 0.88302222155948904756966801319056을(를) 구합니다.

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

분자와 분모 모두에 100000000000000000000000000000000을(를) 곱하여 \frac{1}{0.88302222155948904756966801319056}을(를) 확장합니다.

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

16을(를) 추출 및 상쇄하여 분수 \frac{100000000000000000000000000000000}{88302222155948904756966801319056}을(를) 기약 분수로 약분합니다.

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

11697777844051095979530830215969과(와) 5518888884746806547310425082441의 최소 공배수는 64558736119771056212369977545857700093089334767336350659700329입니다. \frac{600000000000000000000000000000000}{11697777844051095979530830215969} 및 \frac{6250000000000000000000000000000}{5518888884746806547310425082441}을(를) 분모 64558736119771056212369977545857700093089334767336350659700329의 분수로 변환합니다.

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

\frac{3311333330848083928386255049464600000000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} 및 \frac{73111111525319349872067688849806250000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329}의 분모가 같으므로 분자를 빼서 이 둘을 뺍니다.

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

3311333330848083928386255049464600000000000000000000000000000000에서 73111111525319349872067688849806250000000000000000000000000000을(를) 빼고 3238222219322764578514187360614793750000000000000000000000000000을(를) 구합니다.

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

0.3420201433256687의 2제곱을 계산하여 0.11697777844051095979530830215969을(를) 구합니다.

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

64과(와) 0.11697777844051095979530830215969을(를) 곱하여 7.48657782019270142689973133822016(을)를 구합니다.

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

10진수 7.48657782019270142689973133822016을(를) 분수 \frac{11697777844051095979530830215969}{10000000000}(으)로 변환합니다. 1을(를) 추출 및 상쇄하여 분수 \frac{11697777844051095979530830215969}{10000000000}을(를) 기약 분수로 약분합니다.

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

64558736119771056212369977545857700093089334767336350659700329과(와) 1562500000000000000000000000000의 최소 공배수는 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000입니다. \frac{3238222219322764578514187360614793750000000000000000000000000000}{64558736119771056212369977545857700093089334767336350659700329} 및 \frac{11697777844051095979530830215969}{1562500000000000000000000000000}을(를) 분모 100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000의 분수로 변환합니다.

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

\frac{5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000} 및 \frac{755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801}{100873025187142275331828089915402656395452085573963047905781764062500000000000000000000000000}의 분모가 같으므로 분자를 더하여 이 둘을 더합니다.

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

5059722217691819653928417750960615234375000000000000000000000000000000000000000000000000000000과(와) 755193753021799083572852173276890524858698368644990171879009323293911055544430132267890353801을(를) 더하여 5814915970713618737501269924237505759233698368644990171879009323293911055544430132267890353801을(를) 구합니다.

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