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受欢迎的三角函数 >

ln(2+tan((3pi)/(32)))

解答

ln (2 + tan (\frac{3 π}{32}))

解答

ln 2 + 8 2 - 2 + 22 - (2 - 2) (2 - 2 + 2) - 42 2 - 2 - 4 - (2 - 2) (2 - 2 + 2) + 42 2 - 2 + 2 - 8 2 - 2 + 2 + 15 - 82

十进制

0.83436 \dots

求解步骤

ln (2 + tan (\frac{3 π}{32}))

使用三角恒等式改写:

tan (\frac{3 π}{32}) = 8 2 - 2 + 22 - (2 - 2) (2 - 2 + 2) - 42 2 - 2 - 4 - (2 - 2) (2 - 2 + 2) + 42 2 - 2 + 2 - 8 2 - 2 + 2 + 15 - 82

tan (\frac{3 π}{32})

使用三角恒等式改写:

\frac{1 - cos ( \frac{3 π}{16} )}{1 + cos ( \frac{3 π}{16} )}

tan (\frac{3 π}{32})

将

tan (\frac{3 π}{32})

写为

tan (\frac{\frac{3 π}{16}}{2})

= tan (\frac{\frac{3 π}{16}}{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{3 π}{16} )}{1 + cos ( \frac{3 π}{16} )}

= \frac{1 - cos ( \frac{3 π}{16} )}{1 + cos ( \frac{3 π}{16} )}

使用三角恒等式改写:

cos (\frac{3 π}{16}) = \frac{2 + 2 - 2}{2}

cos (\frac{3 π}{16})

使用三角恒等式改写:

\frac{1 + cos ( \frac{3 π}{8} )}{2}

cos (\frac{3 π}{16})

将

cos (\frac{3 π}{16})

写为

cos (\frac{\frac{3 π}{8}}{2})

= cos (\frac{\frac{3 π}{8}}{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{3 π}{8} )}{2}

= \frac{1 + cos ( \frac{3 π}{8} )}{2}

使用三角恒等式改写:

cos (\frac{3 π}{8}) = \frac{2 - 2}{2}

cos (\frac{3 π}{8})

使用三角恒等式改写:

\frac{1 + cos ( \frac{3 π}{4} )}{2}

cos (\frac{3 π}{8})

将

cos (\frac{3 π}{8})

写为

cos (\frac{\frac{3 π}{4}}{2})

= cos (\frac{\frac{3 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{3 π}{4} )}{2}

= \frac{1 + cos ( \frac{3 π}{4} )}{2}

使用以下普通恒等式:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 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}}{2}

化简

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

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

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

\frac{1 + \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}}{2}

使用分式法则:

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

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

数字相乘:

2 \cdot 2 = 4

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

= \frac{2 + 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 - 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}{2}

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

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

化简

\frac{1 - \frac{2 + 2 - 2}{2}}{1 + \frac{2 + 2 - 2}{2}} : 8 2 - 2 + 22 - (2 - 2) (2 - 2 + 2) - 42 2 - 2 - 4 - (2 - 2) (2 - 2 + 2) + 42 2 - 2 + 2 - 8 2 - 2 + 2 + 15 - 82

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

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

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

化简

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

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

数字相乘:

1 \cdot 2 = 2

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

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

化简

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

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

数字相乘:

1 \cdot 2 = 2

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

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

约分:

2

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

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

\frac{2 - 2 + 2 - 2}{2 + 2 + 2 - 2} = 8 2 - 2 + 22 - (2 - 2) (2 - 2 + 2) - 42 2 - 2 - 4 - (2 - 2) (2 - 2 + 2) + 42 2 - 2 + 2 - 8 2 - 2 + 2 + 15 - 82

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

乘以共轭根式

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

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

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

数字相加:

1 + 1 = 2

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

化简

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

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

2^{2} = 4

2^{2}

2^{2} = 4

= 4

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

2 \cdot 2 2 + 2 - 2

数字相乘:

2 \cdot 2 = 4

= 4 2 - 2 + 2

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

使用指数法则:

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

= (2 + 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 - 2

= 4 - 4 2 - 2 + 2 + 2 + 2 - 2

数字相加:

4 + 2 = 6

= 6 + 2 - 2 - 4 2 - 2 + 2

= 6 + 2 - 2 - 4 2 - 2 + 2

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

化简

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

使用根式运算法则:

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

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

使用指数法则:

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

= (2 + 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 - 2

= 4 - (2 - 2 + 2)

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

- (2 + 2 - 2)

打开括号

= - (2) - (2 - 2)

使用加减运算法则

+ (- a) = - a

= - 2 - 2 - 2

= 4 - 2 - 2 - 2

数字相减:

4 - 2 = 2

= - 2 - 2 + 2

= - 2 - 2 + 2

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

乘以共轭根式

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

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

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

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

打开括号

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

使用加减运算法则

+ (- a) = - a

= 6 2 - 2 + 6 \cdot 2 + 2 - 2 2 - 2 + 2 2 - 2 - 4 2 - 2 + 2 2 - 2 - 4 \cdot 2 2 - 2 + 2

化简

6 2 - 2 + 6 \cdot 2 + 2 - 2 2 - 2 + 2 2 - 2 - 4 2 - 2 + 2 2 - 2 - 4 \cdot 2 2 - 2 + 2 : 8 2 - 2 - 4 2 - 2 2 - 2 + 2 - 8 2 - 2 + 2 + 14 - 2

6 2 - 2 + 6 \cdot 2 + 2 - 2 2 - 2 + 2 2 - 2 - 4 2 - 2 + 2 2 - 2 - 4 \cdot 2 2 - 2 + 2

对同类项分组

= 6 2 - 2 + 2 - 2 2 - 2 + 2 2 - 2 - 4 2 - 2 2 - 2 + 2 - 4 \cdot 2 2 - 2 + 2 + 6 \cdot 2

流行的例子

3sin(240)

3 sin (24 0^{\circ})

(4sin(60))/(sin(30))

\frac{4 sin ( 6 0 ^{\circ} )}{sin ( 3 0 ^{\circ} )}

sec(900)

sec (90 0^{\circ})

(600)/(sin(45))

\frac{600}{sin ( 4 5 ^{\circ} )}

sin(4(pi/4))

sin (4 (\frac{π}{4}))