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受欢迎的三角函数 >

sinh(x)=-3/4

解答

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

解答

x = - ln (2)

+1

度数

x = - 39.71440 \dots^{\circ}

求解步骤

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

使用三角恒等式改写

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

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

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

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

化简

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

使用指数运算法则

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

使用指数法则:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 4 = - 6

(e^{x} - (e^{x})^{- 1}) \cdot 4 = - 6

用

e^{x} = u

改写方程式

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

解

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

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

整理后得

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

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}) = - 6

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} = - 6

在两边乘以

u

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

在两边乘以

u

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

化简

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

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

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

解

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

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

将

6 u

para o lado esquerdo

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

两边加上

6 u

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

化简

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

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

改写成标准形式

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

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

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} : \frac{1}{2}

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

数字相加/相减:

- 6 + 10 = 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{4}{8}

约分:

4

= \frac{1}{2}

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

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

数字相减:

- 6 - 10 = - 16

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 16}{8}

使用分式法则:

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

= - \frac{16}{8}

数字相除:

\frac{16}{8} = 2

= - 2

二次方程组的解是:

u = \frac{1}{2}, u = - 2

u = \frac{1}{2}, u = - 2

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{1}{2}, u = - 2

u = \frac{1}{2}, u = - 2

代回

u = e^{x}

，求解

x

解

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

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

使用指数运算法则

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

若

f (x) = g (x)

，则

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

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

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

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

化简

ln (\frac{1}{2}) : - ln (2)

ln (\frac{1}{2})

使用对数计算法则:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (2)

x = - ln (2)

x = - ln (2)

解

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

无解

e^{x} = - 2

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = - ln (2)

x = - ln (2)

作图

流行的例子

tan (3 θ) = 1

7 - 7 sin (θ) = 7

-12cos^2(θ)-8=-17

- 12 cos^{2} (θ) - 8 = - 17

2cos(1/2 x)-sqrt(2)=0

2 cos (\frac{1}{2} x) - 2 = 0

cos (\frac{π}{x}) = 0