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受欢迎的三角函数 >

(25)/(sin(x))=(18)/(sin(36))

解答

\frac{25}{sin ( x )} = \frac{18}{sin ( 3 6 ^{\circ} )}

解答

x = 0.95509 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.95509 \dots + 36 0^{\circ} n

+1

弧度

x = 0.95509 \dots + 2 πn, x = π - 0.95509 \dots + 2 πn

求解步骤

\frac{25}{sin ( x )} = \frac{18}{sin ( 3 6 ^{\circ} )}

sin (3 6^{\circ}) = \frac{2 5 - 5}{4}

sin (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}

两边进行平方

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

利用以下特性:

sin^{2} (x) = 1 - cos^{2} (x)

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

代入

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

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

整理后得

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

在两侧开平方

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

不能为负

sin (3 6^{\circ}) = \frac{5 - 5}{8}

整理后得

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

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

\frac{\frac{5 - 5}{2}}{2}

\frac{5 - 5}{2} = \frac{5 - 5}{2}

\frac{5 - 5}{2}

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}{2}

= \frac{\frac{5 - 5}{2}}{2}

使用分式法则:

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

= \frac{5 - 5}{2 \cdot 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 \cdot 2 2}

2 \cdot 22 = 4

2 \cdot 22

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{25}{sin ( x )} = \frac{18}{\frac{2 5 - 5}{4}}

使用分式交叉相乘

\frac{25}{sin ( x )} = \frac{18}{\frac{2 5 - 5}{4}}

使用分式交叉相乘: 若

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

则

a \cdot d = b \cdot c

25 \cdot \frac{2 5 - 5}{4} = sin (x) \cdot 18

化简

25 \cdot \frac{2 5 - 5}{4} : \frac{25 2 5 - 5}{4}

25 \cdot \frac{2 5 - 5}{4}

分式相乘:

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

= \frac{2 5 - 5 \cdot 25}{4}

分解

4 : 2^{2}

因式分解

4 = 2^{2}

= \frac{25 2 5 - 5}{2 ^{2}}

消掉

\frac{2 5 - 5 \cdot 25}{2 ^{2}} : \frac{25 5 - 5}{2 ^{\frac{3}{2}}}

\frac{2 5 - 5 \cdot 25}{2 ^{2}}

使用根式运算法则:

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

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

= \frac{25 \cdot 2 ^{\frac{1}{2}} 5 - 5}{2 ^{2}}

使用指数法则:

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

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

= \frac{25 5 - 5}{2 ^{2 - \frac{1}{2}}}

数字相减:

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

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

= \frac{25 5 - 5}{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

= \frac{25 5 - 5}{2 2}

\frac{25 5 - 5}{2 2}

有理化:

\frac{25 2 5 - 5}{4}

\frac{25 5 - 5}{2 2}

乘以共轭根式

\frac{2}{2}

= \frac{25 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{25 2 5 - 5}{4}

= \frac{25 2 5 - 5}{4}

\frac{25 2 5 - 5}{4} = sin (x) \cdot 18

\frac{25 2 5 - 5}{4} = sin (x) \cdot 18

交换两边

sin (x) \cdot 18 = \frac{25 2 5 - 5}{4}

两边除以

18

sin (x) \cdot 18 = \frac{25 2 5 - 5}{4}

两边除以

18

\frac{sin ( x ) \cdot 18}{18} = \frac{\frac{25 2 5 - 5}{4}}{18}

化简

\frac{sin ( x ) \cdot 18}{18} = \frac{\frac{25 2 5 - 5}{4}}{18}

化简

\frac{sin ( x ) \cdot 18}{18} : sin (x)

\frac{sin ( x ) \cdot 18}{18}

数字相除:

\frac{18}{18} = 1

= sin (x)

化简

\frac{\frac{25 2 5 - 5}{4}}{18} : \frac{25 2 5 - 5}{72}

\frac{\frac{25 2 5 - 5}{4}}{18}

使用分式法则:

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

= \frac{25 2 5 - 5}{4 \cdot 18}

数字相乘:

4 \cdot 18 = 72

= \frac{25 2 5 - 5}{72}

sin (x) = \frac{25 2 5 - 5}{72}

sin (x) = \frac{25 2 5 - 5}{72}

sin (x) = \frac{25 2 5 - 5}{72}

验证解

找到无定义的点（奇点）:

sin (x) = 0

取

\frac{25}{sin ( x )}

的分母，令其等于零

sin (x) = 0

以下点无定义

sin (x) = 0

将不在定义域的点与解相综合:

sin (x) = \frac{25 2 5 - 5}{72}

使用反三角函数性质

sin (x) = \frac{25 2 5 - 5}{72}

sin (x) = \frac{25 2 5 - 5}{72}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{25 2 5 - 5}{72}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{25 2 5 - 5}{72}) + 36 0^{\circ} n

x = arcsin (\frac{25 2 5 - 5}{72}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{25 2 5 - 5}{72}) + 36 0^{\circ} n

以小数形式表示解

x = 0.95509 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.95509 \dots + 36 0^{\circ} n

作图

流行的例子

7tan(3x)-21tan(x)=0

7 tan (3 x) - 21 tan (x) = 0

cos (A) = - \frac{3}{25}

6sin(2x)=3cos(30)

6 sin (2 x) = 3 cos (3 0^{\circ})

cos (y) = - 1

cot (x) = - \frac{2}{3}