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受欢迎的三角函数 >

sinh(x)= 15/8

解答

sinh (x) = \frac{15}{8}

解答

x = 2 ln (2)

+1

度数

x = 79.42881 \dots^{\circ}

求解步骤

sinh (x) = \frac{15}{8}

使用三角恒等式改写

sinh (x) = \frac{15}{8}

使用双曲函数恒等式:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8}

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8}

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8} : x = 2 ln (2)

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8}

使用分式交叉相乘: 若

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

则

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 8 = 2 \cdot 15

化简

(e^{x} - e^{- x}) \cdot 8 = 30

使用指数运算法则

(e^{x} - e^{- x}) \cdot 8 = 30

使用指数法则:

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

e^{- x} = (e^{x})^{- 1}

(e^{x} - (e^{x})^{- 1}) \cdot 8 = 30

(e^{x} - (e^{x})^{- 1}) \cdot 8 = 30

用

e^{x} = u

改写方程式

(u - (u)^{- 1}) \cdot 8 = 30

解

(u - u^{- 1}) \cdot 8 = 30 : u = 4, u = - \frac{1}{4}

(u - u^{- 1}) \cdot 8 = 30

整理后得

(u - \frac{1}{u}) \cdot 8 = 30

化简

(u - \frac{1}{u}) \cdot 8 : 8 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 8

使用交换律:

(u - \frac{1}{u}) \cdot 8 = 8 (u - \frac{1}{u})

8 (u - \frac{1}{u})

8 (u - \frac{1}{u}) = 30

展开

8 (u - \frac{1}{u}) : 8 u - \frac{8}{u}

8 (u - \frac{1}{u})

使用分配律:

a (b - c) = ab - a c

a = 8, b = u, c = \frac{1}{u}

= 8 u - 8 \cdot \frac{1}{u}

8 \cdot \frac{1}{u} = \frac{8}{u}

8 \cdot \frac{1}{u}

分式相乘:

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

= \frac{1 \cdot 8}{u}

数字相乘:

1 \cdot 8 = 8

= \frac{8}{u}

= 8 u - \frac{8}{u}

8 u - \frac{8}{u} = 30

在两边乘以

u

8 u - \frac{8}{u} = 30

在两边乘以

u

8 uu - \frac{8}{u} u = 30 u

化简

8 uu - \frac{8}{u} u = 30 u

化简

8 uu : 8 u^{2}

8 uu

使用指数法则:

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

uu = u^{1 + 1}

= 8 u^{1 + 1}

数字相加:

1 + 1 = 2

= 8 u^{2}

化简

- \frac{8}{u} u : - 8

- \frac{8}{u} u

分式相乘:

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

= - \frac{8 u}{u}

约分:

u

= - 8

8 u^{2} - 8 = 30 u

8 u^{2} - 8 = 30 u

8 u^{2} - 8 = 30 u

解

8 u^{2} - 8 = 30 u : u = 4, u = - \frac{1}{4}

8 u^{2} - 8 = 30 u

将

30 u

para o lado esquerdo

8 u^{2} - 8 = 30 u

两边减去

30 u

8 u^{2} - 8 - 30 u = 30 u - 30 u

化简

8 u^{2} - 8 - 30 u = 0

8 u^{2} - 8 - 30 u = 0

改写成标准形式

a x^{2} + b x + c = 0

8 u^{2} - 30 u - 8 = 0

使用求根公式求解

8 u^{2} - 30 u - 8 = 0

二次方程求根公式:

若

a = 8, b = - 30, c = - 8

u_{1, 2} = \frac{- ( - 30 ) \pm ( - 30 ) ^{2} - 4 \cdot 8 ( - 8 )}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 30 ) \pm ( - 30 ) ^{2} - 4 \cdot 8 ( - 8 )}{2 \cdot 8}

(- 30)^{2} - 4 \cdot 8 (- 8) = 34

(- 30)^{2} - 4 \cdot 8 (- 8)

使用法则

- (- a) = a

= (- 30)^{2} + 4 \cdot 8 \cdot 8

使用指数法则:

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

若

n

是偶数

(- 30)^{2} = 3 0^{2}

= 3 0^{2} + 4 \cdot 8 \cdot 8

数字相乘:

4 \cdot 8 \cdot 8 = 256

= 3 0^{2} + 256

3 0^{2} = 900

= 900 + 256

数字相加:

900 + 256 = 1156

= 1156

因式分解数字:

1156 = 3 4^{2}

= 3 4^{2}

使用根式运算法则:

n a^{n} = a

3 4^{2} = 34

= 34

u_{1, 2} = \frac{- ( - 30 ) \pm 34}{2 \cdot 8}

将解分隔开

u_{1} = \frac{- ( - 30 ) + 34}{2 \cdot 8}, u_{2} = \frac{- ( - 30 ) - 34}{2 \cdot 8}

u = \frac{- ( - 30 ) + 34}{2 \cdot 8} : 4

\frac{- ( - 30 ) + 34}{2 \cdot 8}

使用法则

- (- a) = a

= \frac{30 + 34}{2 \cdot 8}

数字相加:

30 + 34 = 64

= \frac{64}{2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{64}{16}

数字相除:

\frac{64}{16} = 4

= 4

u = \frac{- ( - 30 ) - 34}{2 \cdot 8} : - \frac{1}{4}

\frac{- ( - 30 ) - 34}{2 \cdot 8}

使用法则

- (- a) = a

= \frac{30 - 34}{2 \cdot 8}

数字相减:

30 - 34 = - 4

= \frac{- 4}{2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 4}{16}

使用分式法则:

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

= - \frac{4}{16}

约分:

4

= - \frac{1}{4}

二次方程组的解是:

u = 4, u = - \frac{1}{4}

u = 4, u = - \frac{1}{4}

验证解

找到无定义的点（奇点）:

u = 0

取

(u - u^{- 1}) 8

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = 4, u = - \frac{1}{4}

u = 4, u = - \frac{1}{4}

代回

u = e^{x}

，求解

x

解

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

e^{x} = 4

使用指数运算法则

e^{x} = 4

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (4)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (4)

化简

ln (4) : 2 ln (2)

ln (4)

改写

4

为幂形式:

4 = 2^{2}

= ln (2^{2})

使用对数计算法则:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (2^{2}) = 2 ln (2)

= 2 ln (2)

x = 2 ln (2)

x = 2 ln (2)

解

e^{x} = - \frac{1}{4} : x \in R

无解

e^{x} = - \frac{1}{4}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = 2 ln (2)

x = 2 ln (2)

作图

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