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受欢迎的三角函数 >

sinh(x)= 5/4

解答

sinh (x) = \frac{5}{4}

解答

x = ln (\frac{5 + 41}{4})

+1

度数

x = 60.02265 \dots^{\circ}

求解步骤

sinh (x) = \frac{5}{4}

使用三角恒等式改写

sinh (x) = \frac{5}{4}

使用双曲函数恒等式:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{4}

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{4}

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{4} : x = ln (\frac{5 + 41}{4})

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{4}

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 4 = 2 \cdot 5

化简

(e^{x} - e^{- x}) \cdot 4 = 10

使用指数运算法则

(e^{x} - e^{- x}) \cdot 4 = 10

使用指数法则:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 4 = 10

(e^{x} - (e^{x})^{- 1}) \cdot 4 = 10

用

e^{x} = u

改写方程式

(u - (u)^{- 1}) \cdot 4 = 10

解

(u - u^{- 1}) \cdot 4 = 10 : u = \frac{5 + 41}{4}, u = \frac{5 - 41}{4}

(u - u^{- 1}) \cdot 4 = 10

整理后得

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

化简

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

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

使用交换律:

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

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

4 (u - \frac{1}{u}) = 10

展开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

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

4 \cdot \frac{1}{u}

分式相乘:

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

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

数字相乘:

1 \cdot 4 = 4

= \frac{4}{u}

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

4 u - \frac{4}{u} = 10

在两边乘以

u

4 u - \frac{4}{u} = 10

在两边乘以

u

4 uu - \frac{4}{u} u = 10 u

化简

4 uu - \frac{4}{u} u = 10 u

化简

4 uu : 4 u^{2}

4 uu

使用指数法则:

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

uu = u^{1 + 1}

= 4 u^{1 + 1}

数字相加:

1 + 1 = 2

= 4 u^{2}

化简

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

- \frac{4}{u} u

分式相乘:

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

= - \frac{4 u}{u}

约分:

u

= - 4

4 u^{2} - 4 = 10 u

4 u^{2} - 4 = 10 u

4 u^{2} - 4 = 10 u

解

4 u^{2} - 4 = 10 u : u = \frac{5 + 41}{4}, u = \frac{5 - 41}{4}

4 u^{2} - 4 = 10 u

将

10 u

para o lado esquerdo

4 u^{2} - 4 = 10 u

两边减去

10 u

4 u^{2} - 4 - 10 u = 10 u - 10 u

化简

4 u^{2} - 4 - 10 u = 0

4 u^{2} - 4 - 10 u = 0

改写成标准形式

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

4 u^{2} - 10 u - 4 = 0

使用求根公式求解

4 u^{2} - 10 u - 4 = 0

二次方程求根公式:

若

a = 4, b = - 10, c = - 4

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

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

(- 10)^{2} - 4 \cdot 4 (- 4) = 241

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

(- 10)^{2} = 1 0^{2}

= 1 0^{2} + 4 \cdot 4 \cdot 4

数字相乘:

4 \cdot 4 \cdot 4 = 64

= 1 0^{2} + 64

1 0^{2} = 100

= 100 + 64

数字相加:

100 + 64 = 164

= 164

164

质因数分解:

2^{2} \cdot 41

164

164

除以

2164 = 82 \cdot 2

= 2 \cdot 82

82

除以

282 = 41 \cdot 2

= 2 \cdot 2 \cdot 41

2, 41

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 41

= 2^{2} \cdot 41

= 2^{2} \cdot 41

使用根式运算法则:

n ab = n a n b

= 41 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 241

u_{1, 2} = \frac{- ( - 10 ) \pm 2 41}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 10 ) + 2 41}{2 \cdot 4}, u_{2} = \frac{- ( - 10 ) - 2 41}{2 \cdot 4}

u = \frac{- ( - 10 ) + 2 41}{2 \cdot 4} : \frac{5 + 41}{4}

\frac{- ( - 10 ) + 2 41}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{10 + 2 41}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{10 + 2 41}{8}

分解

10 + 241 : 2 (5 + 41)

10 + 241

改写为

= 2 \cdot 5 + 241

因式分解出通项

2

= 2 (5 + 41)

= \frac{2 ( 5 + 41 )}{8}

约分:

2

= \frac{5 + 41}{4}

u = \frac{- ( - 10 ) - 2 41}{2 \cdot 4} : \frac{5 - 41}{4}

\frac{- ( - 10 ) - 2 41}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{10 - 2 41}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{10 - 2 41}{8}

分解

10 - 241 : 2 (5 - 41)

10 - 241

改写为

= 2 \cdot 5 - 241

因式分解出通项

2

= 2 (5 - 41)

= \frac{2 ( 5 - 41 )}{8}

约分:

2

= \frac{5 - 41}{4}

二次方程组的解是:

u = \frac{5 + 41}{4}, u = \frac{5 - 41}{4}

u = \frac{5 + 41}{4}, u = \frac{5 - 41}{4}

验证解

找到无定义的点（奇点）:

u = 0

取

(u - u^{- 1}) 4

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 = \frac{5 + 41}{4}, u = \frac{5 - 41}{4}

u = \frac{5 + 41}{4}, u = \frac{5 - 41}{4}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{5 + 41}{4} : x = ln (\frac{5 + 41}{4})

e^{x} = \frac{5 + 41}{4}

使用指数运算法则

e^{x} = \frac{5 + 41}{4}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{5 + 41}{4})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{5 + 41}{4})

x = ln (\frac{5 + 41}{4})

解

e^{x} = \frac{5 - 41}{4} : x \in R

无解

e^{x} = \frac{5 - 41}{4}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = ln (\frac{5 + 41}{4})

x = ln (\frac{5 + 41}{4})

作图

流行的例子

3 - 4 cos (A) = 1

sin (a) = - \frac{3}{5}

cot(x)=cot(3x-50)

cot (x) = cot (3 x - 50)

sin (θ) = \frac{33}{55}

2 cos (x) - 4 = - 3