求解 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)}

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

求值

$$\frac{107911000512775026917055443408919316779890884913062057631792086466736686481041}{33833371706466818736292319830373191000111189050000000000000000000000000000000}\approx 3.189484082$$

求解步骤

对题中的三角函数求值

$$\frac{\frac{2}{2} 0.05233595624294383 \cdot 0.9945218953682733 \cdot 0.9986295347545738}{0.20791169081775931} - \frac{0.45399049973954675 ^ {2} - 0.8910065241883679 ^ {2}}{2 \cdot 0.10452846326765346 \cdot 0.9564956763167232}$$

2 除以 2 得 1。

$$\frac{1\times 0.05233595624294383\times 0.9945218953682733\times 0.9986295347545738}{0.20791169081775931}-\frac{0.45399049973954675^{2}-0.8910065241883679^{2}}{2\times 0.10452846326765346\times 0.9564956763167232}$$

将 1 与 0.05233595624294383 相乘，得到 0.05233595624294383。

$$\frac{0.05233595624294383\times 0.9945218953682733\times 0.9986295347545738}{0.20791169081775931}-\frac{0.45399049973954675^{2}-0.8910065241883679^{2}}{2\times 0.10452846326765346\times 0.9564956763167232}$$

将 0.05233595624294383 与 0.9945218953682733 相乘，得到 0.052049254398643513504402375988739。

$$\frac{0.052049254398643513504402375988739\times 0.9986295347545738}{0.20791169081775931}-\frac{0.45399049973954675^{2}-0.8910065241883679^{2}}{2\times 0.10452846326765346\times 0.9564956763167232}$$

将 0.052049254398643513504402375988739 与 0.9986295347545738 相乘，得到 0.0519779227044398258017752025124954343964775444382。

$$\frac{0.0519779227044398258017752025124954343964775444382}{0.20791169081775931}-\frac{0.45399049973954675^{2}-0.8910065241883679^{2}}{2\times 0.10452846326765346\times 0.9564956763167232}$$

将分子和分母同时乘以 10000000000000000000000000000000000000000000000000 以展开 \frac{0.0519779227044398258017752025124954343964775444382}{0.20791169081775931}。

$$\frac{519779227044398258017752025124954343964775444382}{2079116908177593100000000000000000000000000000000}-\frac{0.45399049973954675^{2}-0.8910065241883679^{2}}{2\times 0.10452846326765346\times 0.9564956763167232}$$

通过求根和消去 2，将分数 \frac{519779227044398258017752025124954343964775444382}{2079116908177593100000000000000000000000000000000} 降低为最简分数。

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{0.45399049973954675^{2}-0.8910065241883679^{2}}{2\times 0.10452846326765346\times 0.9564956763167232}$$

计算 2 的 0.45399049973954675 乘方，得到 0.2061073738537633976795858954355625。

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{0.2061073738537633976795858954355625-0.8910065241883679^{2}}{2\times 0.10452846326765346\times 0.9564956763167232}$$

计算 2 的 0.8910065241883679 乘方，得到 0.79389262614623663165984166575041。

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{0.2061073738537633976795858954355625-0.79389262614623663165984166575041}{2\times 0.10452846326765346\times 0.9564956763167232}$$

将 0.2061073738537633976795858954355625 减去 0.79389262614623663165984166575041，得到 -0.5877852522924732339802557703148475。

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{-0.5877852522924732339802557703148475}{2\times 0.10452846326765346\times 0.9564956763167232}$$

将 2 与 0.10452846326765346 相乘，得到 0.20905692653530692。

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{-0.5877852522924732339802557703148475}{0.20905692653530692\times 0.9564956763167232}$$

将 0.20905692653530692 与 0.9564956763167232 相乘，得到 0.199962046335083909067305240684544。

$$\frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000}-\frac{-0.5877852522924732339802557703148475}{0.199962046335083909067305240684544}$$

将分子和分母同时乘以 10000000000000000000000000000000000 以展开 \frac{-0.5877852522924732339802557703148475}{0.199962046335083909067305240684544}。

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

通过求根和消去 15，将分数 \frac{-5877852522924732339802557703148475}{1999620463350839090673052406845440} 降低为最简分数。

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

-\frac{391856834861648822653503846876565}{133308030890055939378203493789696} 的相反数是 \frac{391856834861648822653503846876565}{133308030890055939378203493789696}。

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

1039558454088796550000000000000000000000000000000 和 133308030890055939378203493789696 的最小公倍数是 33833371706466818736292319830373191000111189050000000000000000000000000000000。将 \frac{259889613522199129008876012562477171982387722191}{1039558454088796550000000000000000000000000000000} 和 \frac{391856834861648822653503846876565}{133308030890055939378203493789696} 转换为带分母 33833371706466818736292319830373191000111189050000000000000000000000000000000 的分数。

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

由于\frac{8458342926616704407721757006270970284559525574500045913042086466736686481041}{33833371706466818736292319830373191000111189050000000000000000000000000000000}和\frac{99452657586158322509333686402648346495331359338562011718750000000000000000000}{33833371706466818736292319830373191000111189050000000000000000000000000000000}具有相同的分母，可通过分子相加来求和。

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

8458342926616704407721757006270970284559525574500045913042086466736686481041 与 99452657586158322509333686402648346495331359338562011718750000000000000000000 相加，得到 107911000512775026917055443408919316779890884913062057631792086466736686481041。

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