求解 2*1divtan(8^{circ})*1divtan(17^circ)*1divtan(45^circ)*1divtan(73^circ)*1divtan(82^circ)

$$2 \times 1 \div \tan ( 8 ^ { \circ } ) \times 1 \div \tan ( 17 ^ { \circ } ) \times 1 \div \tan ( 45 ^ { \circ } ) \times 1 \div \tan ( 73 ^ { \circ } ) \times 1 \div \tan ( 82 ^ { \circ } )$$

求值

$$\frac{50000000000000000000000000000000000000000000000000000000000000}{24999999999999952320918945316322629374650050821000940948202031}\approx 2$$

求解步骤

对题中的三角函数求值

$$2 \cdot 1 / 0.14054083470239145 \cdot 1 / 0.3057306814586604 \cdot 1 / \tan(45) \cdot 1 / 3.2708526184841404 \cdot 1 / 7.115369722384195$$

将 2 与 1 相乘，得到 2。

$$\frac{1\times \frac{1\times \frac{\frac{\frac{2}{0.14054083470239145}\times 1}{0.3057306814586604}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

将分子和分母同时乘以 100000000000000000 以展开 \frac{2}{0.14054083470239145}。

$$\frac{1\times \frac{1\times \frac{\frac{\frac{200000000000000000}{14054083470239145}\times 1}{0.3057306814586604}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

通过求根和消去 5，将分数 \frac{200000000000000000}{14054083470239145} 降低为最简分数。

$$\frac{1\times \frac{1\times \frac{\frac{\frac{40000000000000000}{2810816694047829}\times 1}{0.3057306814586604}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

将 \frac{40000000000000000}{2810816694047829} 与 1 相乘，得到 \frac{40000000000000000}{2810816694047829}。

$$\frac{1\times \frac{1\times \frac{\frac{\frac{40000000000000000}{2810816694047829}}{0.3057306814586604}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

将 \frac{\frac{40000000000000000}{2810816694047829}}{0.3057306814586604} 化为简分数。

$$\frac{1\times \frac{1\times \frac{\frac{40000000000000000}{2810816694047829\times 0.3057306814586604}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

将 2810816694047829 与 0.3057306814586604 相乘，得到 859352903326621.7159602038682716。

$$\frac{1\times \frac{1\times \frac{\frac{40000000000000000}{859352903326621.7159602038682716}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

将分子和分母同时乘以 10000000000000000 以展开 \frac{40000000000000000}{859352903326621.7159602038682716}。

$$\frac{1\times \frac{1\times \frac{\frac{400000000000000000000000000000000}{8593529033266217159602038682716}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

通过求根和消去 4，将分数 \frac{400000000000000000000000000000000}{8593529033266217159602038682716} 降低为最简分数。

$$\frac{1\times \frac{1\times \frac{\frac{100000000000000000000000000000000}{2148382258316554289900509670679}\times 1}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

将 \frac{100000000000000000000000000000000}{2148382258316554289900509670679} 与 1 相乘，得到 \frac{100000000000000000000000000000000}{2148382258316554289900509670679}。

$$\frac{1\times \frac{1\times \frac{\frac{100000000000000000000000000000000}{2148382258316554289900509670679}}{\tan(45)}}{3.2708526184841404}}{7.115369722384195}$$

从三角函数值表中获取 \tan(45) 的值。

$$\frac{1\times \frac{1\times \frac{\frac{100000000000000000000000000000000}{2148382258316554289900509670679}}{1}}{3.2708526184841404}}{7.115369722384195}$$

任何数除以一都等于其本身。

$$\frac{1\times \frac{1\times \frac{100000000000000000000000000000000}{2148382258316554289900509670679}}{3.2708526184841404}}{7.115369722384195}$$

将 1 与 \frac{100000000000000000000000000000000}{2148382258316554289900509670679} 相乘，得到 \frac{100000000000000000000000000000000}{2148382258316554289900509670679}。

$$\frac{1\times \frac{\frac{100000000000000000000000000000000}{2148382258316554289900509670679}}{3.2708526184841404}}{7.115369722384195}$$

将 \frac{\frac{100000000000000000000000000000000}{2148382258316554289900509670679}}{3.2708526184841404} 化为简分数。

$$\frac{1\times \frac{100000000000000000000000000000000}{2148382258316554289900509670679\times 3.2708526184841404}}{7.115369722384195}$$

将 2148382258316554289900509670679 与 3.2708526184841404 相乘，得到 7027041735119572517754492936408.8736997883993316。

$$\frac{1\times \frac{100000000000000000000000000000000}{7027041735119572517754492936408.8736997883993316}}{7.115369722384195}$$

将分子和分母同时乘以 10000000000000000 以展开 \frac{100000000000000000000000000000000}{7027041735119572517754492936408.8736997883993316}。

$$\frac{1\times \frac{1000000000000000000000000000000000000000000000000}{70270417351195725177544929364088736997883993316}}{7.115369722384195}$$

通过求根和消去 4，将分数 \frac{1000000000000000000000000000000000000000000000000}{70270417351195725177544929364088736997883993316} 降低为最简分数。

$$\frac{1\times \frac{250000000000000000000000000000000000000000000000}{17567604337798931294386232341022184249470998329}}{7.115369722384195}$$

将 1 与 \frac{250000000000000000000000000000000000000000000000}{17567604337798931294386232341022184249470998329} 相乘，得到 \frac{250000000000000000000000000000000000000000000000}{17567604337798931294386232341022184249470998329}。

$$\frac{\frac{250000000000000000000000000000000000000000000000}{17567604337798931294386232341022184249470998329}}{7.115369722384195}$$

将 \frac{\frac{250000000000000000000000000000000000000000000000}{17567604337798931294386232341022184249470998329}}{7.115369722384195} 化为简分数。

$$\frac{250000000000000000000000000000000000000000000000}{17567604337798931294386232341022184249470998329\times 7.115369722384195}$$

将 17567604337798931294386232341022184249470998329 与 7.115369722384195 相乘，得到 124999999999999761604594726581613146873250254105.004704741010155。

$$\frac{250000000000000000000000000000000000000000000000}{124999999999999761604594726581613146873250254105.004704741010155}$$

将分子和分母同时乘以 1000000000000000 以展开 \frac{250000000000000000000000000000000000000000000000}{124999999999999761604594726581613146873250254105.004704741010155}。

$$\frac{250000000000000000000000000000000000000000000000000000000000000}{124999999999999761604594726581613146873250254105004704741010155}$$

通过求根和消去 5，将分数 \frac{250000000000000000000000000000000000000000000000000000000000000}{124999999999999761604594726581613146873250254105004704741010155} 降低为最简分数。

$$\frac{50000000000000000000000000000000000000000000000000000000000000}{24999999999999952320918945316322629374650050821000940948202031}$$