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受欢迎的三角函数 >

(8.2)/(sin(72))

解答

\frac{8.2}{sin ( 7 2 ^{\circ} )}

解答

\frac{41 2 ( 5 - 5 ) 5 + 5}{50}

+1

十进制

8.62199 \dots

求解步骤

\frac{8.2}{sin ( 7 2 ^{\circ} )}

= \frac{\frac{41}{5}}{sin ( 7 2 ^{\circ} )}

使用三角恒等式改写:

sin (7 2^{\circ}) = \frac{2 5 + 5}{4}

sin (7 2^{\circ})

使用三角恒等式改写:

cos (1 8^{\circ})

sin (7 2^{\circ})

利用以下特性:

sin (x) = cos (9 0^{\circ} - x)

= cos (9 0^{\circ} - 7 2^{\circ})

化简

= cos (1 8^{\circ})

= cos (1 8^{\circ})

使用三角恒等式改写:

\frac{1 + cos ( 3 6 ^{\circ} )}{2}

cos (1 8^{\circ})

将

cos (1 8^{\circ})

写为

cos (\frac{3 6 ^{\circ}}{2})

= cos (\frac{3 6 ^{\circ}}{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, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] [18 0^{\circ}, 27 0^{\circ}] [27 0^{\circ}, 36 0^{\circ}] 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 ( 3 6 ^{\circ} )}{2}

= \frac{1 + cos ( 3 6 ^{\circ} )}{2}

使用三角恒等式改写:

cos (3 6^{\circ}) = \frac{5 + 1}{4}

cos (3 6^{\circ})

显示:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

使用以下积化和差公式:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

显示:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

利用以下特性:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

两边除以

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代入

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

显示:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

使用因式分解法则:

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

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

整理后得

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

显示:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

利用以下特性:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

两边除以

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代入

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

代入

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

整理后得

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

两边加上

\frac{1}{4}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

整理后得

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

在两侧开平方

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

不能为负

sin (1 8^{\circ})

不能为负

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

以下方程式相加

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

整理后得

cos (3 6^{\circ}) = \frac{5 + 1}{4}

= \frac{5 + 1}{4}

= \frac{1 + \frac{5 + 1}{4}}{2}

化简

\frac{1 + \frac{5 + 1}{4}}{2} : \frac{2 5 + 5}{4}

\frac{1 + \frac{5 + 1}{4}}{2}

\frac{1 + \frac{5 + 1}{4}}{2} = \frac{5 + 5}{8}

\frac{1 + \frac{5 + 1}{4}}{2}

化简

1 + \frac{5 + 1}{4} : \frac{5 + 5}{4}

1 + \frac{5 + 1}{4}

将项转换为分式:

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

= \frac{1 \cdot 4}{4} + \frac{5 + 1}{4}

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 4 + 5 + 1}{4}

1 \cdot 4 + 5 + 1 = 5 + 5

1 \cdot 4 + 5 + 1

数字相乘:

1 \cdot 4 = 4

= 4 + 5 + 1

数字相加:

4 + 1 = 5

= 5 + 5

= \frac{5 + 5}{4}

= \frac{\frac{5 + 5}{4}}{2}

使用分式法则:

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

= \frac{5 + 5}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{5 + 5}{8}

= \frac{5 + 5}{8}

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{5 + 5}{8}

8 = 22

8

8

质因数分解:

2^{3}

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

使用指数法则:

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

= 2^{2} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22

= \frac{5 + 5}{2 2}

\frac{5 + 5}{2 2}

有理化:

\frac{2 5 + 5}{4}

\frac{5 + 5}{2 2}

乘以共轭根式

\frac{2}{2}

= \frac{5 + 5 2}{2 2 2}

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{2 5 + 5}{4}

= \frac{2 5 + 5}{4}

= \frac{2 5 + 5}{4}

= \frac{\frac{41}{5}}{\frac{2 5 + 5}{4}}

化简

\frac{\frac{41}{5}}{\frac{2 5 + 5}{4}} : \frac{41 2 ( 5 - 5 ) 5 + 5}{50}

\frac{\frac{41}{5}}{\frac{2 5 + 5}{4}}

分式相除:

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

= \frac{41 \cdot 4}{5 2 5 + 5}

数字相乘:

41 \cdot 4 = 164

= \frac{164}{5 2 5 + 5}

分解

164 : 2^{2} \cdot 41

因式分解

164 = 2^{2} \cdot 41

= \frac{2 ^{2} \cdot 41}{5 2 5 + 5}

消掉

\frac{2 ^{2} \cdot 41}{5 2 5 + 5} : \frac{41 \cdot 2 ^{\frac{3}{2}}}{5 5 + 5}

\frac{2 ^{2} \cdot 41}{5 2 5 + 5}

使用根式运算法则:

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

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

= \frac{2 ^{2} \cdot 41}{5 \cdot 2 ^{\frac{1}{2}} 5 + 5}

使用指数法则:

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

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

= \frac{41 \cdot 2 ^{- \frac{1}{2} + 2}}{5 5 + 5}

数字相减:

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

= \frac{41 \cdot 2 ^{\frac{3}{2}}}{5 5 + 5}

= \frac{41 \cdot 2 ^{\frac{3}{2}}}{5 5 + 5}

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{41 \cdot 2 2}{5 5 + 5}

数字相乘:

41 \cdot 2 = 82

= \frac{82 2}{5 5 + 5}

\frac{82 2}{5 5 + 5}

有理化:

\frac{41 2 ( 5 - 5 ) 5 + 5}{50}

\frac{82 2}{5 5 + 5}

乘以共轭根式

\frac{5 + 5}{5 + 5}

= \frac{82 2 5 + 5}{5 5 + 5 5 + 5}

5 5 + 5 5 + 5 = 25 + 5^{\frac{3}{2}}

5 5 + 5 5 + 5

使用根式运算法则:

a a = a

5 + 5 5 + 5 = 5 + 5

= 5 (5 + 5)

使用分配律:

a (b + c) = ab + a c

a = 5, b = 5, c = 5

= 5 \cdot 5 + 55

化简

5 \cdot 5 + 55 : 25 + 5^{\frac{3}{2}}

5 \cdot 5 + 55

5 \cdot 5 = 25

5 \cdot 5

数字相乘:

5 \cdot 5 = 25

= 25

55 = 5^{\frac{3}{2}}

55

使用指数法则:

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

55 = 5 \cdot 5^{\frac{1}{2}} = 5^{1 + \frac{1}{2}}

= 5^{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}

= 5^{\frac{3}{2}}

= 25 + 5^{\frac{3}{2}}

= 25 + 5^{\frac{3}{2}}

= \frac{82 2 5 + 5}{25 + 5 ^{\frac{3}{2}}}

5^{\frac{3}{2}} = 55

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

5^{\frac{3}{2}} = 5^{1 + \frac{1}{2}}

= 5^{1 + \frac{1}{2}}

使用指数法则:

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

= 5^{1} \cdot 5^{\frac{1}{2}}

整理后得

= 55

= \frac{82 2 5 + 5}{25 + 5 5}

乘以共轭根式

\frac{25 - 5 5}{25 - 5 5}

= \frac{82 2 5 + 5 ( 25 - 5 5 )}{( 25 + 5 5 ) ( 25 - 5 5 )}

(25 + 55) (25 - 55) = 500

(25 + 55) (25 - 55)

55 = 5^{\frac{3}{2}}

55

使用指数法则:

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

55 = 5 \cdot 5^{\frac{1}{2}} = 5^{1 + \frac{1}{2}}

= 5^{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}

= 5^{\frac{3}{2}}

= (25 + 5^{\frac{3}{2}}) (25 - 55)

55 = 5^{\frac{3}{2}}

55

使用指数法则:

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

55 = 5 \cdot 5^{\frac{1}{2}} = 5^{1 + \frac{1}{2}}

= 5^{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}

= 5^{\frac{3}{2}}

= (25 + 5^{\frac{3}{2}}) (25 - 5^{\frac{3}{2}})

使用平方差公式:

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

a = 25, b = 5^{\frac{3}{2}}

= 2 5^{2} - (5^{\frac{3}{2}})^{2}

化简

2 5^{2} - (5^{\frac{3}{2}})^{2} : 500

2 5^{2} - (5^{\frac{3}{2}})^{2}

2 5^{2} = 625

2 5^{2}

2 5^{2} = 625

= 625

(5^{\frac{3}{2}})^{2} = 125

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

使用指数法则:

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

= 5^{\frac{3}{2} \cdot 2}

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

\frac{3}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 3

= 5^{3}

5^{3} = 125

= 125

= 625 - 125

数字相减:

625 - 125 = 500

= 500

= 500

= \frac{82 2 ( 25 - 5 5 ) 5 + 5}{500}

约分:

2

= \frac{41 2 ( 25 - 5 5 ) 5 + 5}{250}

分解

412 (25 - 55) 5 + 5 : 2052 (5 - 5) 5 + 5

412 (25 - 55) 5 + 5

分解

25 - 55 : 5 (5 - 5)

25 - 55

改写为

= 5 \cdot 5 - 55

因式分解出通项

5

= 5 (5 - 5)

= 412 \cdot 5 (5 - 5) 5 + 5

整理后得

= 2052 (5 - 5) 5 + 5

= \frac{205 2 ( 5 - 5 ) 5 + 5}{250}

约分:

5

= \frac{41 2 ( 5 - 5 ) 5 + 5}{50}

= \frac{41 2 ( 5 - 5 ) 5 + 5}{50}

= \frac{41 2 ( 5 - 5 ) 5 + 5}{50}

流行的例子

sin (3 \cdot \frac{π}{3})

\frac{1}{sin ( 5 0 ^{\circ} )}

(2tan(22.5))/(1-tan^2(22.5))

\frac{2 tan ( 22. 5 ^{\circ} )}{1 - tan ^{2} ( 22. 5 ^{\circ} )}

arctan (\frac{14}{12})

cos^{2} (- 6 0^{\circ})