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受欢迎的三角函数 >

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

解答

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

解答

x = ln (\frac{5 + 3 2}{7}), x = ln (\frac{5 - 3 2}{7})

+1

度数

x = 15.92349 \dots^{\circ}, x = - 127.41593 \dots^{\circ}

求解步骤

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

使用三角恒等式改写

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

使用双曲函数恒等式:

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

4 cosh (x) + 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 5

使用双曲函数恒等式:

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

4 \cdot \frac{e ^{x} + e ^{- x}}{2} + 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 5

4 \cdot \frac{e ^{x} + e ^{- x}}{2} + 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 5

4 \cdot \frac{e ^{x} + e ^{- x}}{2} + 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 5 : x = ln (\frac{5 + 3 2}{7}), x = ln (\frac{5 - 3 2}{7})

4 \cdot \frac{e ^{x} + e ^{- x}}{2} + 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 5

使用指数运算法则

4 \cdot \frac{e ^{x} + e ^{- x}}{2} + 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 5

使用指数法则:

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

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

4 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} + 3 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} = 5

4 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} + 3 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} = 5

用

e^{x} = u

改写方程式

4 \cdot \frac{u + ( u ) ^{- 1}}{2} + 3 \cdot \frac{u - ( u ) ^{- 1}}{2} = 5

解

4 \cdot \frac{u + u ^{- 1}}{2} + 3 \cdot \frac{u - u ^{- 1}}{2} = 5 : u = \frac{5 + 3 2}{7}, u = \frac{5 - 3 2}{7}

4 \cdot \frac{u + u ^{- 1}}{2} + 3 \cdot \frac{u - u ^{- 1}}{2} = 5

整理后得

\frac{2 ( u ^{2} + 1 )}{u} + \frac{3 ( u ^{2} - 1 )}{2 u} = 5

乘以最小公倍数

\frac{2 ( u ^{2} + 1 )}{u} + \frac{3 ( u ^{2} - 1 )}{2 u} = 5

找到

u, 2 u

的最小公倍数:

2 u

u, 2 u

最小公倍数 (LCM)

计算出由出现在

u

或

2 u

中的因子组成的表达式

= 2 u

乘以最小公倍数=

2 u

\frac{2 ( u ^{2} + 1 )}{u} \cdot 2 u + \frac{3 ( u ^{2} - 1 )}{2 u} \cdot 2 u = 5 \cdot 2 u

化简

\frac{2 ( u ^{2} + 1 )}{u} \cdot 2 u + \frac{3 ( u ^{2} - 1 )}{2 u} \cdot 2 u = 5 \cdot 2 u

化简

\frac{2 ( u ^{2} + 1 )}{u} \cdot 2 u : 4 (u^{2} + 1)

\frac{2 ( u ^{2} + 1 )}{u} \cdot 2 u

分式相乘:

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

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

约分:

u

= 2 (u^{2} + 1) \cdot 2

数字相乘:

2 \cdot 2 = 4

= 4 (u^{2} + 1)

化简

\frac{3 ( u ^{2} - 1 )}{2 u} \cdot 2 u : 3 (u^{2} - 1)

\frac{3 ( u ^{2} - 1 )}{2 u} \cdot 2 u

分式相乘:

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

= \frac{3 ( u ^{2} - 1 ) \cdot 2 u}{2 u}

约分:

2

= \frac{3 ( u ^{2} - 1 ) u}{u}

约分:

u

= 3 (u^{2} - 1)

化简

5 \cdot 2 u : 10 u

5 \cdot 2 u

数字相乘:

5 \cdot 2 = 10

= 10 u

4 (u^{2} + 1) + 3 (u^{2} - 1) = 10 u

4 (u^{2} + 1) + 3 (u^{2} - 1) = 10 u

4 (u^{2} + 1) + 3 (u^{2} - 1) = 10 u

解

4 (u^{2} + 1) + 3 (u^{2} - 1) = 10 u : u = \frac{5 + 3 2}{7}, u = \frac{5 - 3 2}{7}

4 (u^{2} + 1) + 3 (u^{2} - 1) = 10 u

展开

4 (u^{2} + 1) + 3 (u^{2} - 1) : 7 u^{2} + 1

4 (u^{2} + 1) + 3 (u^{2} - 1)

乘开

4 (u^{2} + 1) : 4 u^{2} + 4

4 (u^{2} + 1)

使用分配律:

a (b + c) = ab + a c

a = 4, b = u^{2}, c = 1

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

数字相乘:

4 \cdot 1 = 4

= 4 u^{2} + 4

= 4 u^{2} + 4 + 3 (u^{2} - 1)

乘开

3 (u^{2} - 1) : 3 u^{2} - 3

3 (u^{2} - 1)

使用分配律:

a (b - c) = ab - a c

a = 3, b = u^{2}, c = 1

= 3 u^{2} - 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

= 3 u^{2} - 3

= 4 u^{2} + 4 + 3 u^{2} - 3

化简

4 u^{2} + 4 + 3 u^{2} - 3 : 7 u^{2} + 1

4 u^{2} + 4 + 3 u^{2} - 3

对同类项分组

= 4 u^{2} + 3 u^{2} + 4 - 3

同类项相加:

4 u^{2} + 3 u^{2} = 7 u^{2}

= 7 u^{2} + 4 - 3

数字相加/相减:

4 - 3 = 1

= 7 u^{2} + 1

= 7 u^{2} + 1

7 u^{2} + 1 = 10 u

将

10 u

para o lado esquerdo

7 u^{2} + 1 = 10 u

两边减去

10 u

7 u^{2} + 1 - 10 u = 10 u - 10 u

化简

7 u^{2} + 1 - 10 u = 0

7 u^{2} + 1 - 10 u = 0

改写成标准形式

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

7 u^{2} - 10 u + 1 = 0

使用求根公式求解

7 u^{2} - 10 u + 1 = 0

二次方程求根公式:

若

a = 7, b = - 10, c = 1

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 7 \cdot 1}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 7 \cdot 1}{2 \cdot 7}

(- 10)^{2} - 4 \cdot 7 \cdot 1 = 62

(- 10)^{2} - 4 \cdot 7 \cdot 1

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 7 \cdot 1

数字相乘:

4 \cdot 7 \cdot 1 = 28

= 1 0^{2} - 28

1 0^{2} = 100

= 100 - 28

数字相减:

100 - 28 = 72

= 72

72

质因数分解:

2^{3} \cdot 3^{2}

72

72

除以

272 = 36 \cdot 2

= 2 \cdot 36

36

除以

236 = 18 \cdot 2

= 2 \cdot 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

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

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

使用指数法则:

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

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

使用根式运算法则:

= 2 2^{2} 3^{2}

使用根式运算法则:

2^{2} = 2

= 22 3^{2}

使用根式运算法则:

3^{2} = 3

= 2 \cdot 32

整理后得

= 62

u_{1, 2} = \frac{- ( - 10 ) \pm 6 2}{2 \cdot 7}

将解分隔开

u_{1} = \frac{- ( - 10 ) + 6 2}{2 \cdot 7}, u_{2} = \frac{- ( - 10 ) - 6 2}{2 \cdot 7}

u = \frac{- ( - 10 ) + 6 2}{2 \cdot 7} : \frac{5 + 3 2}{7}

\frac{- ( - 10 ) + 6 2}{2 \cdot 7}

使用法则

- (- a) = a

= \frac{10 + 6 2}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{10 + 6 2}{14}

分解

10 + 62 : 2 (5 + 32)

10 + 62

改写为

= 2 \cdot 5 + 2 \cdot 32

因式分解出通项

2

= 2 (5 + 32)

= \frac{2 ( 5 + 3 2 )}{14}

约分:

2

= \frac{5 + 3 2}{7}

u = \frac{- ( - 10 ) - 6 2}{2 \cdot 7} : \frac{5 - 3 2}{7}

\frac{- ( - 10 ) - 6 2}{2 \cdot 7}

使用法则

- (- a) = a

= \frac{10 - 6 2}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{10 - 6 2}{14}

分解

10 - 62 : 2 (5 - 32)

10 - 62

改写为

= 2 \cdot 5 - 2 \cdot 32

因式分解出通项

2

= 2 (5 - 32)

= \frac{2 ( 5 - 3 2 )}{14}

约分:

2

= \frac{5 - 3 2}{7}

二次方程组的解是:

u = \frac{5 + 3 2}{7}, u = \frac{5 - 3 2}{7}

u = \frac{5 + 3 2}{7}, u = \frac{5 - 3 2}{7}

验证解

找到无定义的点（奇点）:

u = 0

取

4 \frac{u + u ^{- 1}}{2} + 3 \frac{u - u ^{- 1}}{2}

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 + 3 2}{7}, u = \frac{5 - 3 2}{7}

u = \frac{5 + 3 2}{7}, u = \frac{5 - 3 2}{7}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{5 + 3 2}{7} : x = ln (\frac{5 + 3 2}{7})

e^{x} = \frac{5 + 3 2}{7}

使用指数运算法则

e^{x} = \frac{5 + 3 2}{7}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{5 + 3 2}{7})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{5 + 3 2}{7})

x = ln (\frac{5 + 3 2}{7})

解

e^{x} = \frac{5 - 3 2}{7} : x = ln (\frac{5 - 3 2}{7})

e^{x} = \frac{5 - 3 2}{7}

使用指数运算法则

e^{x} = \frac{5 - 3 2}{7}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{5 - 3 2}{7})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{5 - 3 2}{7})

x = ln (\frac{5 - 3 2}{7})

x = ln (\frac{5 + 3 2}{7}), x = ln (\frac{5 - 3 2}{7})

x = ln (\frac{5 + 3 2}{7}), x = ln (\frac{5 - 3 2}{7})

作图

流行的例子

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