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受欢迎的三角函数 >

cos(arctan(5/7)+arctan(1/6))

解答

cos (arctan (\frac{5}{7}) + arctan (\frac{1}{6}))

解答

\frac{2}{2}

+1

十进制

0.70710 \dots

求解步骤

cos (arctan (\frac{5}{7}) + arctan (\frac{1}{6}))

使用三角恒等式改写:

arctan (\frac{5}{7}) + arctan (\frac{1}{6}) = arctan (1)

arctan (\frac{5}{7}) + arctan (\frac{1}{6})

使用和差化积恒等式:

arctan (s) + arctan (t) = arctan (\frac{s + t}{1 - s t})

= arctan (\frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{7} \cdot \frac{1}{6}})

化简:

\frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{7} \cdot \frac{1}{6}} = 1

\frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{7} \cdot \frac{1}{6}}

\frac{5}{7} \cdot \frac{1}{6} = \frac{5}{42}

\frac{5}{7} \cdot \frac{1}{6}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{5 \cdot 1}{7 \cdot 6}

数字相乘:

5 \cdot 1 = 5

= \frac{5}{7 \cdot 6}

数字相乘:

7 \cdot 6 = 42

= \frac{5}{42}

= \frac{\frac{5}{7} + \frac{1}{6}}{1 - \frac{5}{42}}

化简

\frac{5}{7} + \frac{1}{6} : \frac{37}{42}

\frac{5}{7} + \frac{1}{6}

7, 6

的最小公倍数:

42

7, 6

最小公倍数 (LCM)

7

质因数分解:

7

7

7

是质数，因此无法因数分解

= 7

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

将每个因子乘以它在

7

或

6

中出现的最多次数

= 7 \cdot 2 \cdot 3

数字相乘:

7 \cdot 2 \cdot 3 = 42

= 42

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

42

对于

\frac{5}{7} :

将分母和分子乘以

6

\frac{5}{7} = \frac{5 \cdot 6}{7 \cdot 6} = \frac{30}{42}

对于

\frac{1}{6} :

将分母和分子乘以

7

\frac{1}{6} = \frac{1 \cdot 7}{6 \cdot 7} = \frac{7}{42}

= \frac{30}{42} + \frac{7}{42}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{30 + 7}{42}

数字相加:

30 + 7 = 37

= \frac{37}{42}

= \frac{\frac{37}{42}}{1 - \frac{5}{42}}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{37}{42 ( 1 - \frac{5}{42} )}

化简

1 - \frac{5}{42} : \frac{37}{42}

1 - \frac{5}{42}

将项转换为分式:

1 = \frac{1 \cdot 42}{42}

= \frac{1 \cdot 42}{42} - \frac{5}{42}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 42 - 5}{42}

1 \cdot 42 - 5 = 37

1 \cdot 42 - 5

数字相乘:

1 \cdot 42 = 42

= 42 - 5

数字相减:

42 - 5 = 37

= 37

= \frac{37}{42}

= \frac{37}{42 \cdot \frac{37}{42}}

乘

42 \cdot \frac{37}{42} : 37

42 \cdot \frac{37}{42}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{37 \cdot 42}{42}

约分:

42

= 37

= \frac{37}{37}

使用法则

\frac{a}{a} = 1

= 1

= arctan (1)

= cos (arctan (1))

使用三角恒等式改写:

cos (arctan (1)) = \frac{1 + 1 ^{2}}{1 + 1 ^{2}}

利用以下特性:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

= \frac{1 + 1 ^{2}}{1 + 1 ^{2}}

= \frac{1 + 1 ^{2}}{1 + 1 ^{2}}

化简

= \frac{2}{2}

流行的例子

arctan (1.75)

sqrt((1-cos(150))/2)

\frac{1 - cos ( 15 0 ^{\circ} )}{2}

arctan (\frac{7}{9})

tan (63 0^{\circ})

arctan (\frac{- 4}{4})