zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

sinh(4x)= 3/4

解答

sinh (4 x) = \frac{3}{4}

解答

x = \frac{1}{4} ln (2)

+1

度数

x = 9.92860 \dots^{\circ}

求解步骤

sinh (4 x) = \frac{3}{4}

使用三角恒等式改写

sinh (4 x) = \frac{3}{4}

使用双曲函数恒等式:

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

\frac{e ^{4 x} - e ^{- 4 x}}{2} = \frac{3}{4}

\frac{e ^{4 x} - e ^{- 4 x}}{2} = \frac{3}{4}

\frac{e ^{4 x} - e ^{- 4 x}}{2} = \frac{3}{4} : x = \frac{1}{4} ln (2)

\frac{e ^{4 x} - e ^{- 4 x}}{2} = \frac{3}{4}

使用分式交叉相乘: 若

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

则

a \cdot d = b \cdot c

(e^{4 x} - e^{- 4 x}) \cdot 4 = 2 \cdot 3

化简

(e^{4 x} - e^{- 4 x}) \cdot 4 = 6

使用指数运算法则

(e^{4 x} - e^{- 4 x}) \cdot 4 = 6

使用指数法则:

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

e^{4 x} = (e^{x})^{4}, e^{- 4 x} = (e^{x})^{- 4}

((e^{x})^{4} - (e^{x})^{- 4}) \cdot 4 = 6

((e^{x})^{4} - (e^{x})^{- 4}) \cdot 4 = 6

用

e^{x} = u

改写方程式

((u)^{4} - (u)^{- 4}) \cdot 4 = 6

解

(u^{4} - u^{- 4}) \cdot 4 = 6 : u = 2, u = - 2

(u^{4} - u^{- 4}) \cdot 4 = 6

整理后得

(u^{4} - \frac{1}{u ^{4}}) \cdot 4 = 6

化简

(u^{4} - \frac{1}{u ^{4}}) \cdot 4 : 4 (u^{4} - \frac{1}{u ^{4}})

(u^{4} - \frac{1}{u ^{4}}) \cdot 4

使用交换律:

(u^{4} - \frac{1}{u ^{4}}) \cdot 4 = 4 (u^{4} - \frac{1}{u ^{4}})

4 (u^{4} - \frac{1}{u ^{4}})

4 (u^{4} - \frac{1}{u ^{4}}) = 6

展开

4 (u^{4} - \frac{1}{u ^{4}}) : 4 u^{4} - \frac{4}{u ^{4}}

4 (u^{4} - \frac{1}{u ^{4}})

使用分配律:

a (b - c) = ab - a c

a = 4, b = u^{4}, c = \frac{1}{u ^{4}}

= 4 u^{4} - 4 \cdot \frac{1}{u ^{4}}

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

4 \cdot \frac{1}{u ^{4}}

分式相乘:

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

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

数字相乘:

1 \cdot 4 = 4

= \frac{4}{u ^{4}}

= 4 u^{4} - \frac{4}{u ^{4}}

4 u^{4} - \frac{4}{u ^{4}} = 6

在两边乘以

u^{4}

4 u^{4} - \frac{4}{u ^{4}} = 6

在两边乘以

u^{4}

4 u^{4} u^{4} - \frac{4}{u ^{4}} u^{4} = 6 u^{4}

化简

4 u^{4} u^{4} - \frac{4}{u ^{4}} u^{4} = 6 u^{4}

化简

4 u^{4} u^{4} : 4 u^{8}

4 u^{4} u^{4}

使用指数法则:

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

u^{4} u^{4} = u^{4 + 4}

= 4 u^{4 + 4}

数字相加:

4 + 4 = 8

= 4 u^{8}

化简

- \frac{4}{u ^{4}} u^{4} : - 4

- \frac{4}{u ^{4}} u^{4}

分式相乘:

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

= - \frac{4 u ^{4}}{u ^{4}}

约分:

u^{4}

= - 4

4 u^{8} - 4 = 6 u^{4}

4 u^{8} - 4 = 6 u^{4}

4 u^{8} - 4 = 6 u^{4}

解

4 u^{8} - 4 = 6 u^{4} : u = 2, u = - 2

4 u^{8} - 4 = 6 u^{4}

将

6 u^{4}

para o lado esquerdo

4 u^{8} - 4 = 6 u^{4}

两边减去

6 u^{4}

4 u^{8} - 4 - 6 u^{4} = 6 u^{4} - 6 u^{4}

化简

4 u^{8} - 4 - 6 u^{4} = 0

4 u^{8} - 4 - 6 u^{4} = 0

改写成标准形式

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

4 u^{8} - 6 u^{4} - 4 = 0

用

v = u^{2}, v^{2} = u^{4}

和

v^{4} = u^{8}

改写方程式

4 v^{4} - 6 v^{2} - 4 = 0

解

4 v^{4} - 6 v^{2} - 4 = 0 : v = 2, v = - 2

4 v^{4} - 6 v^{2} - 4 = 0

用

u = v^{2}

和

u^{2} = v^{4}

改写方程式

4 u^{2} - 6 u - 4 = 0

解

4 u^{2} - 6 u - 4 = 0 : u = 2, u = - \frac{1}{2}

4 u^{2} - 6 u - 4 = 0

使用求根公式求解

4 u^{2} - 6 u - 4 = 0

二次方程求根公式:

若

a = 4, b = - 6, c = - 4

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 4 ( - 4 )}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 4 ( - 4 )}{2 \cdot 4}

(- 6)^{2} - 4 \cdot 4 (- 4) = 10

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 4 \cdot 4 = 64

= 6^{2} + 64

6^{2} = 36

= 36 + 64

数字相加:

36 + 64 = 100

= 100

因式分解数字:

100 = 1 0^{2}

= 1 0^{2}

使用根式运算法则:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- ( - 6 ) \pm 10}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 6 ) + 10}{2 \cdot 4}, u_{2} = \frac{- ( - 6 ) - 10}{2 \cdot 4}

u = \frac{- ( - 6 ) + 10}{2 \cdot 4} : 2

\frac{- ( - 6 ) + 10}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{6 + 10}{2 \cdot 4}

数字相加:

6 + 10 = 16

= \frac{16}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{16}{8}

数字相除:

\frac{16}{8} = 2

= 2

u = \frac{- ( - 6 ) - 10}{2 \cdot 4} : - \frac{1}{2}

\frac{- ( - 6 ) - 10}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{6 - 10}{2 \cdot 4}

数字相减:

6 - 10 = - 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 4}{8}

使用分式法则:

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

= - \frac{4}{8}

约分:

4

= - \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^{2}

，求解

v

解

v^{2} = 2 : v = 2, v = - 2

v^{2} = 2

对于

x^{2} = f (a)

解为

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

v = 2, v = - 2

解

v^{2} = - \frac{1}{2} : v \in R

无解

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

x^{2}

在

x

内不能为负

\in R

v \in R 无解

解为

v = 2, v = - 2

v = 2, v = - 2

代回

v = u^{2}

，求解

u

解

u^{2} = 2 : u = 2, u = - 2

u^{2} = 2

对于

x^{2} = f (a)

解为

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

u = 2, u = - 2

解

u^{2} = - 2 : u \in R

无解

u^{2} = - 2

x^{2}

在

x

内不能为负

\in R

u \in R 无解

解为

u = 2, u = - 2

u = 2, u = - 2

验证解

找到无定义的点（奇点）:

u = 0

取

(u^{4} - u^{- 4}) 4

的分母，令其等于零

解

u^{4} = 0 : u = 0

u^{4} = 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:

u = 2, u = - 2

u = 2, u = - 2

代回

u = e^{x}

，求解

x

解

e^{x} = 2 : x = \frac{1}{4} ln (2)

e^{x} = 2

使用指数运算法则

e^{x} = 2

使用指数法则:

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

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

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

使用指数法则:

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

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

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

若

f (x) = g (x)

，则

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

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

使用对数计算法则:

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

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

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

化简

x = \frac{1}{4} ln (2)

x = \frac{1}{4} ln (2)

解

e^{x} = - 2 : x \in R

无解

e^{x} = - 2

使用指数运算法则

e^{x} = - 2

使用指数法则:

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

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

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

e^{x} = - 2^{\frac{1}{2} \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}{4} ln (2)

x = \frac{1}{4} ln (2)

作图

流行的例子

tan(x)+sec(x)=3

tan (x) + sec (x) = 3

2sin(pi/3-x)-1=0

2 sin (\frac{π}{3} - x) - 1 = 0

cos(θ)=-(11)/(sqrt(170))

cos (θ) = - \frac{11}{170}

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

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

tan((3x)/2+pi/2)=1

tan (\frac{3 x}{2} + \frac{π}{2}) = 1