Resolver frac{2/2sin3^{circ}sin84^{circ}sin87^{circ}}{sin12^{circ}}-frac{sin^{2}27^{circ}-cos^{2}27^{circ}}{2sin6^circcos6^circ(3-4sin^212^circ)}

$$\frac { \frac { 2 } { 2 } \sin 3 ^ { \circ } \sin 84 ^ { \circ } \sin 87 ^ { \circ } } { \sin 12 ^ { \circ } } - \frac { \sin ^ { 2 } 27 ^ { \circ } - \cos ^ { 2 } 27 ^ { \circ } } { 2 \sin 6 ^ { \circ } \cos 6 ^ { \circ } ( 3 - 4 \sin ^ { 2 } 12 ^ { \circ } ) }$$

Avaliar

$$\frac{107911000512775026917055443408919316779890884913062057631792086466736686481041}{33833371706466818736292319830373191000111189050000000000000000000000000000000}\approx 3,189484082$$

Passos Para a Resolução

Evaluate trigonometric functions in the problem

$$\frac{\frac{2}{2} 0,05233595624294383 \cdot 0,9945218953682733 \cdot 0,9986295347545738}{0,20791169081775931} - \frac{0,45399049973954675 ^ {2} - 0,8910065241883679 ^ {2}}{2 \cdot 0,10452846326765346 \cdot 0,9564956763167232}$$

Dividir 2 por 2 para obter 1.

$$\frac{1\times 0,05233595624294383\times 0,9945218953682733\times 0,9986295347545738}{0,20791169081775931}-\frac{0,45399049973954675^{2}-0,8910065241883679^{2}}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Multiplique 1 e 0,05233595624294383 para obter 0,05233595624294383.

$$\frac{0,05233595624294383\times 0,9945218953682733\times 0,9986295347545738}{0,20791169081775931}-\frac{0,45399049973954675^{2}-0,8910065241883679^{2}}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Multiplique 0,05233595624294383 e 0,9945218953682733 para obter 0,052049254398643513504402375988739.

$$\frac{0,052049254398643513504402375988739\times 0,9986295347545738}{0,20791169081775931}-\frac{0,45399049973954675^{2}-0,8910065241883679^{2}}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Multiplique 0,052049254398643513504402375988739 e 0,9986295347545738 para obter 0,0519779227044398258017752025124954343964775444382.

$$\frac{0,0519779227044398258017752025124954343964775444382}{0,20791169081775931}-\frac{0,45399049973954675^{2}-0,8910065241883679^{2}}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Expanda \frac{0,0519779227044398258017752025124954343964775444382}{0,20791169081775931} ao multiplicar o numerador e o denominador por 10000000000000000000000000000000000000000000000000.

$$\frac{519779227044398258017752025124954343964775444382}{2079116908177593100000000000000000000000000000000}-\frac{0,45399049973954675^{2}-0,8910065241883679^{2}}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Reduza a fração \frac{519779227044398258017752025124954343964775444382}{2079116908177593100000000000000000000000000000000} para os termos mais baixos ao retirar e anular 2.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{0,45399049973954675^{2}-0,8910065241883679^{2}}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Calcule 0,45399049973954675 elevado a 2 e obtenha 0,2061073738537633976795858954355625.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{0,2061073738537633976795858954355625-0,8910065241883679^{2}}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Calcule 0,8910065241883679 elevado a 2 e obtenha 0,79389262614623663165984166575041.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{0,2061073738537633976795858954355625-0,79389262614623663165984166575041}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Subtraia 0,79389262614623663165984166575041 de 0,2061073738537633976795858954355625 para obter -0,5877852522924732339802557703148475.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{-0,5877852522924732339802557703148475}{2\times 0,10452846326765346\times 0,9564956763167232}$$

Multiplique 2 e 0,10452846326765346 para obter 0,20905692653530692.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{-0,5877852522924732339802557703148475}{0,20905692653530692\times 0,9564956763167232}$$

Multiplique 0,20905692653530692 e 0,9564956763167232 para obter 0,199962046335083909067305240684544.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{-0,5877852522924732339802557703148475}{0,199962046335083909067305240684544}$$

Expanda \frac{-0,5877852522924732339802557703148475}{0,199962046335083909067305240684544} ao multiplicar o numerador e o denominador por 10000000000000000000000000000000000.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{-5877852522924732339802557703148475}{1999620463350839090673052406845440}$$

Reduza a fração \frac{-5877852522924732339802557703148475}{1999620463350839090673052406845440} para os termos mais baixos ao retirar e anular 15.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\left(-\frac{391856834861648822653503846876565}{133308030890055939378203493789696}\right)$$

O oposto de -\frac{391856834861648822653503846876565}{133308030890055939378203493789696} é \frac{391856834861648822653503846876565}{133308030890055939378203493789696}.

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}+\frac{391856834861648822653503846876565}{133308030890055939378203493789696}$$

O mínimo múltiplo comum de 1039558454088796550000000000000000000000000000000 e 133308030890055939378203493789696 é 33833371706466818736292319830373191000111189050000000000000000000000000000000. Converta \frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000} e \frac{391856834861648822653503846876565}{133308030890055939378203493789696} em frações com o denominador 33833371706466818736292319830373191000111189050000000000000000000000000000000.

$$\frac{8458342926616704407721757006270970284559525574500045913042086466736686481041}{33833371706466818736292319830373191000111189050000000000000000000000000000000}+\frac{99452657586158322509333686402648346495331359338562011718750000000000000000000}{33833371706466818736292319830373191000111189050000000000000000000000000000000}$$

Uma vez que \frac{8458342926616704407721757006270970284559525574500045913042086466736686481041}{33833371706466818736292319830373191000111189050000000000000000000000000000000} e \frac{99452657586158322509333686402648346495331359338562011718750000000000000000000}{33833371706466818736292319830373191000111189050000000000000000000000000000000} têm o mesmo denominador, some-os ao somar os respetivos numeradores.

$$\frac{8458342926616704407721757006270970284559525574500045913042086466736686481041+99452657586158322509333686402648346495331359338562011718750000000000000000000}{33833371706466818736292319830373191000111189050000000000000000000000000000000}$$

Some 8458342926616704407721757006270970284559525574500045913042086466736686481041 e 99452657586158322509333686402648346495331359338562011718750000000000000000000 para obter 107911000512775026917055443408919316779890884913062057631792086466736686481041.

$$\frac{107911000512775026917055443408919316779890884913062057631792086466736686481041}{33833371706466818736292319830373191000111189050000000000000000000000000000000}$$