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受欢迎的三角函数 >

cot(pi/(16))

解答

cot (\frac{π}{16})

解答

\frac{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

+1

十进制

5.02733 \dots

求解步骤

cot (\frac{π}{16})

使用三角恒等式改写:

\frac{1}{tan ( \frac{π}{16} )}

cot (\frac{π}{16})

使用基本三角恒等式:

cot (x) = \frac{1}{tan ( x )}

= \frac{1}{tan ( \frac{π}{16} )}

= \frac{1}{tan ( \frac{π}{16} )}

使用三角恒等式改写:

tan (\frac{π}{16}) = - 4 2 + 2 - 22 2 + 2 + 7 + 42

tan (\frac{π}{16})

使用三角恒等式改写:

\frac{1 - cos ( \frac{π}{8} )}{1 + cos ( \frac{π}{8} )}

tan (\frac{π}{16})

将

tan (\frac{π}{16})

写为

tan (\frac{\frac{π}{8}}{2})

= tan (\frac{\frac{π}{8}}{2})

使用半角公式:

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

使用三角恒等式改写:

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

利用以下特性

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

两边进行平方

tan^{2} (θ) = \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

使用三角恒等式改写:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

使用倍角公式

cos (2 θ) = 1 - 2 sin^{2} (θ)

交换两边

2 sin^{2} (θ) - 1 = - cos (2 θ)

两边加上

1

2 sin^{2} (θ) = 1 - cos (2 θ)

两边除以

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

使用三角恒等式改写:

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

使用倍角公式

cos (2 θ) = 2 cos^{2} (θ) - 1

交换两边

2 cos^{2} (θ) - 1 = cos (2 θ)

两边加上

1

2 sin^{2} (θ) = 1 + cos (2 θ)

两边除以

2

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

tan^{2} (θ) = \frac{\frac{1 - c o s ( 2 θ )}{2}}{\frac{1 + c o s ( 2 θ )}{2}}

化简

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

用

\frac{θ}{2}

替代

θ

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( 2 \cdot \frac{θ}{2} )}{1 + cos ( 2 \cdot \frac{θ}{2} )}

化简

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

:

range [0, \frac{π}{2}] [\frac{π}{2}, π] quadrant I II tan positive negative

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

= \frac{1 - cos ( \frac{π}{8} )}{1 + cos ( \frac{π}{8} )}

= \frac{1 - cos ( \frac{π}{8} )}{1 + cos ( \frac{π}{8} )}

使用三角恒等式改写:

cos (\frac{π}{8}) = \frac{2 + 2}{2}

cos (\frac{π}{8})

使用三角恒等式改写:

\frac{1 + cos ( \frac{π}{4} )}{2}

cos (\frac{π}{8})

将

cos (\frac{π}{8})

写为

cos (\frac{\frac{π}{4}}{2})

= cos (\frac{\frac{π}{4}}{2})

使用半角公式:

cos (\frac{θ}{2}) = \frac{1 + cos ( θ )}{2}

使用倍角公式

cos (2 θ) = 2 cos^{2} (θ) - 1

用

\frac{θ}{2}

替代

θ

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

交换两边

2 cos^{2} (\frac{θ}{2}) = 1 + cos (θ)

两边除以

2

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

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

:

range [0, \frac{π}{2}] [\frac{π}{2}, π] [π, \frac{3 π}{2}] [\frac{3 π}{2}, 2 π] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

cos (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

= \frac{1 + cos ( \frac{π}{4} )}{2}

= \frac{1 + cos ( \frac{π}{4} )}{2}

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

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

化简

\frac{1 + \frac{2}{2}}{2} : \frac{2 + 2}{2}

\frac{1 + \frac{2}{2}}{2}

\frac{1 + \frac{2}{2}}{2} = \frac{2 + 2}{4}

\frac{1 + \frac{2}{2}}{2}

化简

1 + \frac{2}{2} : \frac{2 + 2}{2}

1 + \frac{2}{2}

将项转换为分式:

1 = \frac{1 \cdot 2}{2}

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

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 2 + 2}{2}

数字相乘:

1 \cdot 2 = 2

= \frac{2 + 2}{2}

= \frac{\frac{2 + 2}{2}}{2}

使用分式法则:

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

= \frac{2 + 2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2 + 2}{4}

= \frac{2 + 2}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 + 2}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{2 + 2}{2}

= \frac{2 + 2}{2}

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

化简

\frac{1 - \frac{2 + 2}{2}}{1 + \frac{2 + 2}{2}} : - 4 2 + 2 - 22 2 + 2 + 7 + 42

\frac{1 - \frac{2 + 2}{2}}{1 + \frac{2 + 2}{2}}

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

\frac{1 - \frac{2 + 2}{2}}{1 + \frac{2 + 2}{2}}

化简

1 + \frac{2 + 2}{2} : \frac{2 + 2 + 2}{2}

1 + \frac{2 + 2}{2}

将项转换为分式:

1 = \frac{1 \cdot 2}{2}

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

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 2 + 2 + 2}{2}

数字相乘:

1 \cdot 2 = 2

= \frac{2 + 2 + 2}{2}

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

化简

1 - \frac{2 + 2}{2} : \frac{2 - 2 + 2}{2}

1 - \frac{2 + 2}{2}

将项转换为分式:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{2 + 2}{2}

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 2 - 2 + 2}{2}

数字相乘:

1 \cdot 2 = 2

= \frac{2 - 2 + 2}{2}

= \frac{\frac{2 - 2 + 2}{2}}{\frac{2 + 2 + 2}{2}}

分式相除:

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

= \frac{( 2 - 2 + 2 ) \cdot 2}{2 ( 2 + 2 + 2 )}

约分:

2

= \frac{2 - 2 + 2}{2 + 2 + 2}

= \frac{2 - 2 + 2}{2 + 2 + 2}

\frac{2 - 2 + 2}{2 + 2 + 2} = - 4 2 + 2 - 22 2 + 2 + 7 + 42

\frac{2 - 2 + 2}{2 + 2 + 2}

乘以共轭根式

\frac{2 - 2 + 2}{2 - 2 + 2}

= \frac{( 2 - 2 + 2 ) ( 2 - 2 + 2 )}{( 2 + 2 + 2 ) ( 2 - 2 + 2 )}

(2 - 2 + 2) (2 - 2 + 2) = - 4 2 + 2 + 6 + 2

(2 - 2 + 2) (2 - 2 + 2)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

(2 - 2 + 2) (2 - 2 + 2) = (2 - 2 + 2)^{1 + 1}

= (2 - 2 + 2)^{1 + 1}

数字相加:

1 + 1 = 2

= (2 - 2 + 2)^{2}

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2, b = 2 + 2

= 2^{2} - 2 \cdot 2 2 + 2 + (2 + 2)^{2}

化简

2^{2} - 2 \cdot 2 2 + 2 + (2 + 2)^{2} : - 4 2 + 2 + 6 + 2

2^{2} - 2 \cdot 2 2 + 2 + (2 + 2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 2 + 2 = 4 2 + 2

2 \cdot 2 2 + 2

数字相乘:

2 \cdot 2 = 4

= 4 2 + 2

(2 + 2)^{2} = 2 + 2

(2 + 2)^{2}

使用根式运算法则:

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

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

使用指数法则:

(a^{b})^{c} = a^{b c}

= (2 + 2)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 2 + 2

= 4 - 4 2 + 2 + 2 + 2

数字相加:

4 + 2 = 6

= - 4 2 + 2 + 6 + 2

= - 4 2 + 2 + 6 + 2

(2 + 2 + 2) (2 - 2 + 2) = 2 - 2

(2 + 2 + 2) (2 - 2 + 2)

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = 2, b = 2 + 2

= 2^{2} - (2 + 2)^{2}

化简

2^{2} - (2 + 2)^{2} : 2 - 2

2^{2} - (2 + 2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(2 + 2)^{2} = 2 + 2

(2 + 2)^{2}

使用根式运算法则:

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

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

使用指数法则:

(a^{b})^{c} = a^{b c}

= (2 + 2)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 2 + 2

= 4 - (2 + 2)

- (2 + 2) : - 2 - 2

- (2 + 2)

打开括号

= - (2) - (2)

使用加减运算法则

+ (- a) = - a

= - 2 - 2

= 4 - 2 - 2

数字相减:

4 - 2 = 2

= 2 - 2

= 2 - 2

= \frac{- 4 2 + 2 + 6 + 2}{2 - 2}

乘以共轭根式

\frac{2 + 2}{2 + 2}

= \frac{( - 4 2 + 2 + 6 + 2 ) ( 2 + 2 )}{( 2 - 2 ) ( 2 + 2 )}

(- 4 2 + 2 + 6 + 2) (2 + 2) = - 8 2 + 2 - 42 2 + 2 + 14 + 82

(- 4 2 + 2 + 6 + 2) (2 + 2)

打开括号

= (- 4 2 + 2) \cdot 2 + (- 4 2 + 2) 2 + 6 \cdot 2 + 62 + 2 \cdot 2 + 22

使用加减运算法则

+ (- a) = - a

= - 4 \cdot 2 2 + 2 - 42 2 + 2 + 6 \cdot 2 + 62 + 22 + 22

化简

- 4 \cdot 2 2 + 2 - 42 2 + 2 + 6 \cdot 2 + 62 + 22 + 22 : - 8 2 + 2 - 42 2 + 2 + 14 + 82

- 4 \cdot 2 2 + 2 - 42 2 + 2 + 6 \cdot 2 + 62 + 22 + 22

同类项相加:

62 + 22 = 82

= - 4 \cdot 2 2 + 2 - 42 2 + 2 + 6 \cdot 2 + 82 + 22

数字相乘:

4 \cdot 2 = 8

= - 8 2 + 2 - 42 2 + 2 + 6 \cdot 2 + 82 + 22

数字相乘:

6 \cdot 2 = 12

= - 8 2 + 2 - 42 2 + 2 + 12 + 82 + 22

使用根式运算法则:

a a = a

22 = 2

= - 8 2 + 2 - 42 2 + 2 + 12 + 82 + 2

数字相加:

12 + 2 = 14

= - 8 2 + 2 - 42 2 + 2 + 14 + 82

= - 8 2 + 2 - 42 2 + 2 + 14 + 82

(2 - 2) (2 + 2) = 2

(2 - 2) (2 + 2)

使用平方差公式:

(a - b) (a + b) = a^{2} - b^{2}

a = 2, b = 2

= 2^{2} - (2)^{2}

化简

2^{2} - (2)^{2} : 2

2^{2} - (2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(2)^{2} = 2

(2)^{2}

使用根式运算法则:

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

= (2^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 2

= 4 - 2

数字相减:

4 - 2 = 2

= 2

= 2

= \frac{- 8 2 + 2 - 4 2 2 + 2 + 14 + 8 2}{2}

分解

- 8 2 + 2 - 42 2 + 2 + 14 + 82 : 2 (- 4 2 + 2 - 22 2 + 2 + 7 + 42)

- 8 2 + 2 - 42 2 + 2 + 14 + 82

改写为

= - 2 \cdot 4 2 + 2 - 2 \cdot 22 2 + 2 + 2 \cdot 7 + 2 \cdot 42

因式分解出通项

2

= 2 (- 4 2 + 2 - 22 2 + 2 + 7 + 42)

= \frac{2 ( - 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2 )}{2}

数字相除:

\frac{2}{2} = 1

= - 4 2 + 2 - 22 2 + 2 + 7 + 42

= - 4 2 + 2 - 22 2 + 2 + 7 + 42

= - 4 2 + 2 - 22 2 + 2 + 7 + 42

= \frac{1}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

化简

\frac{1}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2} : \frac{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

\frac{1}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

乘以共轭根式

\frac{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

= \frac{1 \cdot - 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2 - 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

1 \cdot - 4 2 + 2 - 22 2 + 2 + 7 + 42 = - 4 2 + 2 - 22 2 + 2 + 7 + 42

- 4 2 + 2 - 22 2 + 2 + 7 + 42 - 4 2 + 2 - 22 2 + 2 + 7 + 42 = - 4 2 + 2 - 2^{\frac{3}{2}} 2 + 2 + 7 + 42

- 4 2 + 2 - 22 2 + 2 + 7 + 42 - 4 2 + 2 - 22 2 + 2 + 7 + 42

使用根式运算法则:

a a = a

- 4 2 + 2 - 22 2 + 2 + 7 + 42 - 4 2 + 2 - 22 2 + 2 + 7 + 42 = - 4 2 + 2 - 22 2 + 2 + 7 + 42

= - 4 2 + 2 - 22 2 + 2 + 7 + 42

22 2 + 2 = 2^{\frac{3}{2}} 2 + 2

22 2 + 2

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

22 = 2 \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2}}

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

2^{1 + \frac{1}{2}} = 2^{\frac{3}{2}}

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

化简

1 + \frac{1}{2} : \frac{3}{2}

1 + \frac{1}{2}

将项转换为分式:

1 = \frac{1 \cdot 2}{2}

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

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 2 + 1}{2}

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

数字相乘:

1 \cdot 2 = 2

= 2 + 1

数字相加:

2 + 1 = 3

= 3

= \frac{3}{2}

= 2^{\frac{3}{2}}

= 2^{\frac{3}{2}} 2 + 2

= - 4 2 + 2 - 2^{\frac{3}{2}} 2 + 2 + 7 + 42

= \frac{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}{- 4 2 + 2 - 2 ^{\frac{3}{2}} 2 + 2 + 7 + 4 2}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

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

使用指数法则:

x^{a + b} = x^{a} x^{b}

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

整理后得

= 22

= \frac{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

= \frac{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}{- 4 2 + 2 - 2 2 2 + 2 + 7 + 4 2}

流行的例子

cos(60)+sin(45)

cos (6 0^{\circ}) + sin (4 5^{\circ})

cos (\frac{8 π}{6})

cot (\frac{27 π}{2})

arctan(tan((7pi)/9))

arctan (tan (\frac{7 π}{9}))

cos(2arcsin(8/17))

cos (2 arcsin (\frac{8}{17}))