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受欢迎的三角函数 >

cosh(x)=4

解答

cosh (x) = 4

解答

x = ln (4 + 15), x = ln (4 - 15)

+1

度数

x = 118.22623 \dots^{\circ}, x = - 118.22623 \dots^{\circ}

求解步骤

cosh (x) = 4

使用三角恒等式改写

cosh (x) = 4

使用双曲函数恒等式:

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

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

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

\frac{e ^{x} + e ^{- x}}{2} = 4 : x = ln (4 + 15), x = ln (4 - 15)

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

在两边乘以

2

\frac{e ^{x} + e ^{- x}}{2} \cdot 2 = 4 \cdot 2

化简

e^{x} + e^{- x} = 8

使用指数运算法则

e^{x} + e^{- x} = 8

使用指数法则:

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

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

e^{x} + (e^{x})^{- 1} = 8

e^{x} + (e^{x})^{- 1} = 8

用

e^{x} = u

改写方程式

u + (u)^{- 1} = 8

解

u + u^{- 1} = 8 : u = 4 + 15, u = 4 - 15

u + u^{- 1} = 8

整理后得

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

在两边乘以

u

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

在两边乘以

u

uu + \frac{1}{u} u = 8 u

化简

uu + \frac{1}{u} u = 8 u

化简

uu : u^{2}

uu

使用指数法则:

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

uu = u^{1 + 1}

= u^{1 + 1}

数字相加:

1 + 1 = 2

= u^{2}

化简

\frac{1}{u} u : 1

\frac{1}{u} u

分式相乘:

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

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

约分:

u

= 1

u^{2} + 1 = 8 u

u^{2} + 1 = 8 u

u^{2} + 1 = 8 u

解

u^{2} + 1 = 8 u : u = 4 + 15, u = 4 - 15

u^{2} + 1 = 8 u

将

8 u

para o lado esquerdo

u^{2} + 1 = 8 u

两边减去

8 u

u^{2} + 1 - 8 u = 8 u - 8 u

化简

u^{2} + 1 - 8 u = 0

u^{2} + 1 - 8 u = 0

改写成标准形式

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

u^{2} - 8 u + 1 = 0

使用求根公式求解

u^{2} - 8 u + 1 = 0

二次方程求根公式:

若

a = 1, b = - 8, c = 1

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

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

(- 8)^{2} - 4 \cdot 1 \cdot 1 = 215

(- 8)^{2} - 4 \cdot 1 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 8^{2} - 4

8^{2} = 64

= 64 - 4

数字相减:

64 - 4 = 60

= 60

60

质因数分解:

2^{2} \cdot 3 \cdot 5

60

60

除以

260 = 30 \cdot 2

= 2 \cdot 30

30

除以

230 = 15 \cdot 2

= 2 \cdot 2 \cdot 15

15

除以

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 5

2, 3, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 5

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

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

使用根式运算法则:

n ab = n a n b

= 2^{2} 3 \cdot 5

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 3 \cdot 5

整理后得

= 215

u_{1, 2} = \frac{- ( - 8 ) \pm 2 15}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - 8 ) + 2 15}{2 \cdot 1}, u_{2} = \frac{- ( - 8 ) - 2 15}{2 \cdot 1}

u = \frac{- ( - 8 ) + 2 15}{2 \cdot 1} : 4 + 15

\frac{- ( - 8 ) + 2 15}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{8 + 2 15}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8 + 2 15}{2}

分解

8 + 215 : 2 (4 + 15)

8 + 215

改写为

= 2 \cdot 4 + 215

因式分解出通项

2

= 2 (4 + 15)

= \frac{2 ( 4 + 15 )}{2}

数字相除:

\frac{2}{2} = 1

= 4 + 15

u = \frac{- ( - 8 ) - 2 15}{2 \cdot 1} : 4 - 15

\frac{- ( - 8 ) - 2 15}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{8 - 2 15}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8 - 2 15}{2}

分解

8 - 215 : 2 (4 - 15)

8 - 215

改写为

= 2 \cdot 4 - 215

因式分解出通项

2

= 2 (4 - 15)

= \frac{2 ( 4 - 15 )}{2}

数字相除:

\frac{2}{2} = 1

= 4 - 15

二次方程组的解是:

u = 4 + 15, u = 4 - 15

u = 4 + 15, u = 4 - 15

验证解

找到无定义的点（奇点）:

u = 0

取

u + u^{- 1}

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 + 15, u = 4 - 15

u = 4 + 15, u = 4 - 15

代回

u = e^{x}

，求解

x

解

e^{x} = 4 + 15 : x = ln (4 + 15)

e^{x} = 4 + 15

使用指数运算法则

e^{x} = 4 + 15

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (4 + 15)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (4 + 15)

x = ln (4 + 15)

解

e^{x} = 4 - 15 : x = ln (4 - 15)

e^{x} = 4 - 15

使用指数运算法则

e^{x} = 4 - 15

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (4 - 15)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (4 - 15)

x = ln (4 - 15)

x = ln (4 + 15), x = ln (4 - 15)

x = ln (4 + 15), x = ln (4 - 15)

作图

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