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受欢迎的三角函数 >

[3(cos(pi/(12))+sin(pi/(12)))]^3

解答

[3 (cos (\frac{π}{12}) + sin (\frac{π}{12}))]^{3}

解答

\frac{81 6}{4}

+1

十进制

49.60216 \dots

求解步骤

[3 (cos (\frac{π}{12}) + sin (\frac{π}{12}))]^{3}

= (3 (cos (\frac{π}{12}) + sin (\frac{π}{12})))^{3}

使用三角恒等式改写:

cos (\frac{π}{12}) = \frac{6 + 2}{4}

cos (\frac{π}{12})

使用三角恒等式改写:

cos (\frac{π}{4}) cos (\frac{π}{6}) + sin (\frac{π}{4}) sin (\frac{π}{6})

cos (\frac{π}{12})

将

cos (\frac{π}{12})

写为

cos (\frac{π}{4} - \frac{π}{6})

= cos (\frac{π}{4} - \frac{π}{6})

使用角差恒等式:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= cos (\frac{π}{4}) cos (\frac{π}{6}) + sin (\frac{π}{4}) sin (\frac{π}{6})

= cos (\frac{π}{4}) cos (\frac{π}{6}) + sin (\frac{π}{4}) sin (\frac{π}{6})

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}

使用以下普通恒等式:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

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{3}{2}

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

使用以下普通恒等式:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

化简

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} : \frac{6 + 2}{4}

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

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

分式相乘:

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

= \frac{2 3}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2 3}{4}

化简

23 : 6

23

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

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

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

分式相乘:

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

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

乘以:

2 \cdot 1 = 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

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

使用法则

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

= \frac{6 + 2}{4}

= \frac{6 + 2}{4}

使用三角恒等式改写:

sin (\frac{π}{12}) = \frac{6 - 2}{4}

sin (\frac{π}{12})

使用三角恒等式改写:

sin (\frac{π}{4}) cos (\frac{π}{6}) - cos (\frac{π}{4}) sin (\frac{π}{6})

sin (\frac{π}{12})

将

sin (\frac{π}{12})

写为

sin (\frac{π}{4} - \frac{π}{6})

= sin (\frac{π}{4} - \frac{π}{6})

使用角差恒等式:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (\frac{π}{4}) cos (\frac{π}{6}) - cos (\frac{π}{4}) sin (\frac{π}{6})

= sin (\frac{π}{4}) cos (\frac{π}{6}) - cos (\frac{π}{4}) sin (\frac{π}{6})

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

使用以下普通恒等式:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

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{3}{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}

使用以下普通恒等式:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

化简

\frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2} : \frac{6 - 2}{4}

\frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

分式相乘:

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

= \frac{2 3}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2 3}{4}

化简

23 : 6

23

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

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

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

分式相乘:

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

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

乘以:

2 \cdot 1 = 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

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

使用法则

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

= \frac{6 - 2}{4}

= \frac{6 - 2}{4}

= (3 (\frac{6 + 2}{4} + \frac{6 - 2}{4}))^{3}

化简

(3 (\frac{6 + 2}{4} + \frac{6 - 2}{4}))^{3} : \frac{81 6}{4}

(3 (\frac{6 + 2}{4} + \frac{6 - 2}{4}))^{3}

合并分式

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

使用法则

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

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

6 + 2 + 6 - 2 = 26

6 + 2 + 6 - 2

同类项相加:

2 - 2 = 0

= 6 + 6

同类项相加:

6 + 6 = 26

= 26

= \frac{2 6}{4}

约分:

2

= \frac{6}{2}

= (3 (\frac{6}{2}))^{3}

去除括号:

(a) = a

= (3 \cdot \frac{6}{2})^{3}

乘

3 \cdot \frac{6}{2} : \frac{3 3}{2}

3 \cdot \frac{6}{2}

分式相乘:

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

= \frac{6 \cdot 3}{2}

分解

6 : 23

因式分解

6 = 2 \cdot 3

= 2 \cdot 3

使用根式运算法则:

n ab = n a n b

= 23

= \frac{3 2 3}{2}

消掉

\frac{2 3 \cdot 3}{2} : \frac{3 \cdot 3}{2}

\frac{2 3 \cdot 3}{2}

使用根式运算法则:

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

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

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

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

数字相减:

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

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

使用根式运算法则:

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

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

= \frac{3 3}{2}

= \frac{3 \cdot 3}{2}

= (\frac{3 \cdot 3}{2})^{3}

使用指数法则:

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

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

(2)^{3} : 2^{\frac{3}{2}}

使用根式运算法则:

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

= (2^{\frac{1}{2}})^{3}

使用指数法则:

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

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

\frac{1}{2} \cdot 3 = \frac{3}{2}

\frac{1}{2} \cdot 3

分式相乘:

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

= \frac{1 \cdot 3}{2}

数字相乘:

1 \cdot 3 = 3

= \frac{3}{2}

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

= \frac{( 3 \cdot 3 ) ^{3}}{2 ^{\frac{3}{2}}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(33)^{3} = 3^{3} (3)^{3}

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

= 22

(3)^{3} : 3^{\frac{3}{2}}

使用根式运算法则:

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

= (3^{\frac{1}{2}})^{3}

使用指数法则:

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

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

\frac{1}{2} \cdot 3 = \frac{3}{2}

\frac{1}{2} \cdot 3

分式相乘:

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

= \frac{1 \cdot 3}{2}

数字相乘:

1 \cdot 3 = 3

= \frac{3}{2}

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

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

3^{\frac{3}{2}} = 33

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

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

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

使用指数法则:

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

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

整理后得

= 33

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

33 \cdot 3^{3} = 3^{4} 3

33 \cdot 3^{3}

使用指数法则:

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

3 \cdot 3^{3} = 3^{1 + 3}

= 3 \cdot 3^{1 + 3}

数字相加:

1 + 3 = 4

= 3 \cdot 3^{4}

= \frac{3 ^{4} 3}{2 2}

3^{4} = 81

= \frac{81 3}{2 2}

\frac{81 3}{2 2}

有理化:

\frac{81 6}{4}

\frac{81 3}{2 2}

乘以共轭根式

\frac{2}{2}

= \frac{3 \cdot 81 2}{2 2 2}

3 \cdot 812 = 816

3 \cdot 812

使用根式运算法则:

a b = a \cdot b

32 = 3 \cdot 2

= 81 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 816

222 = 4

222

使用指数法则:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

同类项相加:

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

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

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

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

数字相加:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{81 6}{4}

= \frac{81 6}{4}

= \frac{81 6}{4}

流行的例子

π - sin (π)

arctan (- 0.75)

3 cos (2)

ln (sec (0))

e^{π} cos (π)