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受欢迎的三角函数 >

sinh(x40)= 30/40

解答

sinh (x 40) = \frac{30}{40}

解答

x = \frac{1}{40} ln (2)

+1

度数

x = 0.99286 \dots^{\circ}

求解步骤

sinh (x \cdot 40) = \frac{30}{40}

使用三角恒等式改写

sinh (x \cdot 40) = \frac{30}{40}

使用双曲函数恒等式:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40}

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40}

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40} : x = \frac{1}{40} ln (2)

\frac{e ^{x \cdot 40} - e ^{- x \cdot 40}}{2} = \frac{30}{40}

使用分式交叉相乘: 若

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

则

a \cdot d = b \cdot c

(e^{x \cdot 40} - e^{- x \cdot 40}) \cdot 40 = 2 \cdot 30

化简

(e^{x \cdot 40} - e^{- x \cdot 40}) \cdot 40 = 60

使用指数运算法则

(e^{x \cdot 40} - e^{- x \cdot 40}) \cdot 40 = 60

使用指数法则:

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

e^{x 40} = (e^{x})^{40}, e^{- x 40} = (e^{x})^{- 40}

((e^{x})^{40} - (e^{x})^{- 40}) \cdot 40 = 60

((e^{x})^{40} - (e^{x})^{- 40}) \cdot 40 = 60

用

e^{x} = u

改写方程式

((u)^{40} - (u)^{- 40}) \cdot 40 = 60

解

(u^{40} - u^{- 40}) \cdot 40 = 60

整理后得

(u^{40} - \frac{1}{u ^{40}}) \cdot 40 = 60

化简

(u^{40} - \frac{1}{u ^{40}}) \cdot 40 : 40 (u^{40} - \frac{1}{u ^{40}})

(u^{40} - \frac{1}{u ^{40}}) \cdot 40

使用交换律:

(u^{40} - \frac{1}{u ^{40}}) \cdot 40 = 40 (u^{40} - \frac{1}{u ^{40}})

40 (u^{40} - \frac{1}{u ^{40}})

40 (u^{40} - \frac{1}{u ^{40}}) = 60

展开

40 (u^{40} - \frac{1}{u ^{40}}) : 40 u^{40} - \frac{40}{u ^{40}}

40 (u^{40} - \frac{1}{u ^{40}})

使用分配律:

a (b - c) = ab - a c

a = 40, b = u^{40}, c = \frac{1}{u ^{40}}

= 40 u^{40} - 40 \cdot \frac{1}{u ^{40}}

40 \cdot \frac{1}{u ^{40}} = \frac{40}{u ^{40}}

40 \cdot \frac{1}{u ^{40}}

分式相乘:

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

= \frac{1 \cdot 40}{u ^{40}}

数字相乘:

1 \cdot 40 = 40

= \frac{40}{u ^{40}}

= 40 u^{40} - \frac{40}{u ^{40}}

40 u^{40} - \frac{40}{u ^{40}} = 60

在两边乘以

u^{40}

40 u^{40} - \frac{40}{u ^{40}} = 60

在两边乘以

u^{40}

40 u^{40} u^{40} - \frac{40}{u ^{40}} u^{40} = 60 u^{40}

化简

40 u^{40} u^{40} - \frac{40}{u ^{40}} u^{40} = 60 u^{40}

化简

40 u^{40} u^{40} : 40 u^{80}

40 u^{40} u^{40}

使用指数法则:

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

u^{40} u^{40} = u^{40 + 40}

= 40 u^{40 + 40}

数字相加:

40 + 40 = 80

= 40 u^{80}

化简

- \frac{40}{u ^{40}} u^{40} : - 40

- \frac{40}{u ^{40}} u^{40}

分式相乘:

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

= - \frac{40 u ^{40}}{u ^{40}}

约分:

u^{40}

= - 40

40 u^{80} - 40 = 60 u^{40}

40 u^{80} - 40 = 60 u^{40}

40 u^{80} - 40 = 60 u^{40}

解

40 u^{80} - 40 = 60 u^{40}

将

60 u^{40}

para o lado esquerdo

40 u^{80} - 40 = 60 u^{40}

两边减去

60 u^{40}

40 u^{80} - 40 - 60 u^{40} = 60 u^{40} - 60 u^{40}

化简

40 u^{80} - 40 - 60 u^{40} = 0

40 u^{80} - 40 - 60 u^{40} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

40 u^{80} - 60 u^{40} - 40 = 0

用

v = u^{2}, v^{20} = u^{40}

和

v^{40} = u^{80}

改写方程式

40 v^{40} - 60 v^{20} - 40 = 0

解

40 v^{40} - 60 v^{20} - 40 = 0

用

u = v^{2}, u^{10} = v^{20}

和

u^{20} = v^{40}

改写方程式

40 u^{20} - 60 u^{10} - 40 = 0

解

40 u^{20} - 60 u^{10} - 40 = 0

用

v = u^{2}, v^{5} = u^{10}

和

v^{10} = u^{20}

改写方程式

40 v^{10} - 60 v^{5} - 40 = 0

解

40 v^{10} - 60 v^{5} - 40 = 0

用

u = v^{5}

和

u^{2} = v^{10}

改写方程式

40 u^{2} - 60 u - 40 = 0

解

40 u^{2} - 60 u - 40 = 0 : u = 2, u = - \frac{1}{2}

40 u^{2} - 60 u - 40 = 0

使用求根公式求解

40 u^{2} - 60 u - 40 = 0

二次方程求根公式:

若

a = 40, b = - 60, c = - 40

u_{1, 2} = \frac{- ( - 60 ) \pm ( - 60 ) ^{2} - 4 \cdot 40 ( - 40 )}{2 \cdot 40}

u_{1, 2} = \frac{- ( - 60 ) \pm ( - 60 ) ^{2} - 4 \cdot 40 ( - 40 )}{2 \cdot 40}

(- 60)^{2} - 4 \cdot 40 (- 40) = 100

(- 60)^{2} - 4 \cdot 40 (- 40)

使用法则

- (- a) = a

= (- 60)^{2} + 4 \cdot 40 \cdot 40

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 60)^{2} = 6 0^{2}

= 6 0^{2} + 4 \cdot 40 \cdot 40

数字相乘:

4 \cdot 40 \cdot 40 = 6400

= 6 0^{2} + 6400

6 0^{2} = 3600

= 3600 + 6400

数字相加:

3600 + 6400 = 10000

= 10000

因式分解数字:

10000 = 10 0^{2}

= 10 0^{2}

使用根式运算法则:

10 0^{2} = 100

= 100

u_{1, 2} = \frac{- ( - 60 ) \pm 100}{2 \cdot 40}

将解分隔开

u_{1} = \frac{- ( - 60 ) + 100}{2 \cdot 40}, u_{2} = \frac{- ( - 60 ) - 100}{2 \cdot 40}

u = \frac{- ( - 60 ) + 100}{2 \cdot 40} : 2

\frac{- ( - 60 ) + 100}{2 \cdot 40}

使用法则

- (- a) = a

= \frac{60 + 100}{2 \cdot 40}

数字相加:

60 + 100 = 160

= \frac{160}{2 \cdot 40}

数字相乘:

2 \cdot 40 = 80

= \frac{160}{80}

数字相除:

\frac{160}{80} = 2

= 2

u = \frac{- ( - 60 ) - 100}{2 \cdot 40} : - \frac{1}{2}

\frac{- ( - 60 ) - 100}{2 \cdot 40}

使用法则

- (- a) = a

= \frac{60 - 100}{2 \cdot 40}

数字相减:

60 - 100 = - 40

= \frac{- 40}{2 \cdot 40}

数字相乘:

2 \cdot 40 = 80

= \frac{- 40}{80}

使用分式法则:

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

= - \frac{40}{80}

约分:

40

= - \frac{1}{2}

二次方程组的解是:

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

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

代回

u = v^{5}

，求解

v

解

v^{5} = 2

对于

x^{n} = f (a)

，n 为奇数，解为

解

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

对于

x^{n} = f (a)

，n 为奇数，解为

使用根式运算法则:

若

n

是奇数

使用根式运算法则:

使用根式运算法则:

解为

代回

v = u^{2}

，求解

u

解

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

解

无解

x^{2}

在

x

内不能为负

\in R

u \in R 无解

解为

代回

u = v^{2}

，求解

v

解

化简

使用根式运算法则:

= (2^{\frac{1}{5}})^{\frac{1}{2}}

使用指数法则:

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

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

\frac{1}{5} \cdot \frac{1}{2} = \frac{1}{10}

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

数字相乘:

5 \cdot 2 = 10

= \frac{1}{10}

= 2^{\frac{1}{10}}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

解

无解

化简

使用根式运算法则:

= (2^{\frac{1}{5}})^{\frac{1}{2}}

使用指数法则:

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

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

\frac{1}{5} \cdot \frac{1}{2} = \frac{1}{10}

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

数字相乘:

5 \cdot 2 = 10

= \frac{1}{10}

= 2^{\frac{1}{10}}

x^{2}

在

x

内不能为负

\in R

v \in R 无解

解为

代回

v = u^{2}

，求解

u

解

化简

使用根式运算法则:

= (2^{\frac{1}{10}})^{\frac{1}{2}}

使用指数法则:

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

= 2^{\frac{1}{10} \cdot \frac{1}{2}}

\frac{1}{10} \cdot \frac{1}{2} = \frac{1}{20}

\frac{1}{10} \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 1}{10 \cdot 2}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{10 \cdot 2}

数字相乘:

10 \cdot 2 = 20

= \frac{1}{20}

= 2^{\frac{1}{20}}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

解

无解

化简

使用根式运算法则:

= (2^{\frac{1}{10}})^{\frac{1}{2}}

使用指数法则:

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

= 2^{\frac{1}{10} \cdot \frac{1}{2}}

\frac{1}{10} \cdot \frac{1}{2} = \frac{1}{20}

\frac{1}{10} \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 1}{10 \cdot 2}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{10 \cdot 2}

数字相乘:

10 \cdot 2 = 20

= \frac{1}{20}

= 2^{\frac{1}{20}}

x^{2}

在

x

内不能为负

\in R

u \in R 无解

解为

验证解

找到无定义的点（奇点）:

u = 0

取

(u^{40} - u^{- 40}) 40

的分母，令其等于零

解

u^{40} = 0 : u = 0

u^{40} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

代回

u = e^{x}

，求解

x

解

使用指数运算法则

使用指数法则:

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

使用指数法则:

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

e^{x} = 2^{\frac{1}{20} \cdot \frac{1}{2}}

若

f (x) = g (x)

，则

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (2^{\frac{1}{20} \cdot \frac{1}{2}})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (2^{\frac{1}{20} \cdot \frac{1}{2}})

使用对数计算法则:

ln (x^{a}) = a \cdot ln (x)

ln (2^{\frac{1}{20} \cdot \frac{1}{2}}) = \frac{1}{20} \cdot \frac{1}{2} ln (2)

x = \frac{1}{20} \cdot \frac{1}{2} ln (2)

化简

x = \frac{1}{40} ln (2)

x = \frac{1}{40} ln (2)

解

无解

使用指数运算法则

使用指数法则:

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

e^{x} = - 2^{\frac{1}{20} \cdot \frac{1}{2}}

e^{x} = - 2^{\frac{1}{20} \cdot \frac{1}{2}}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = \frac{1}{40} ln (2)

x = \frac{1}{40} ln (2)

作图

流行的例子

4cosh(x)+3sinh(x)=5

5-7sin(x)=2cos^2(x)