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受欢迎的三角函数 >

tan(arcsin(cos(x)))=-sqrt(3)

解答

tan (arcsin (cos (x))) = - 3

解答

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

求解步骤

tan (arcsin (cos (x))) = - 3

两边减去

- 3

tan (arcsin (cos (x))) + 3 = 0

使用三角恒等式改写

3 + tan (arcsin (cos (x)))

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= 3 + \frac{sin ( arcsin ( cos ( x ) ) )}{cos ( arcsin ( cos ( x ) ) )}

化简

3 + \frac{sin ( arcsin ( cos ( x ) ) )}{cos ( arcsin ( cos ( x ) ) )} : 3 + \frac{cos ( x )}{1 - cos ^{2} ( x )}

3 + \frac{sin ( arcsin ( cos ( x ) ) )}{cos ( arcsin ( cos ( x ) ) )}

利用以下特性:

cos (arcsin (x)) = 1 - x^{2}

= 3 + \frac{sin ( arcsin ( cos ( x ) ) )}{1 - cos ^{2} ( x )}

利用以下特性:

sin (arcsin (x)) = x

= 3 + \frac{cos ( x )}{1 - cos ^{2} ( x )}

= 3 + \frac{cos ( x )}{1 - cos ^{2} ( x )}

\frac{cos ( x )}{1 - cos ^{2} ( x )} + 3 = 0

用替代法求解

\frac{cos ( x )}{1 - cos ^{2} ( x )} + 3 = 0

令

: cos (x) = u

\frac{u}{1 - u ^{2}} + 3 = 0

\frac{u}{1 - u ^{2}} + 3 = 0 : u = - \frac{3}{2}

\frac{u}{1 - u ^{2}} + 3 = 0

在两边乘以

1 - u^{2}

\frac{u}{1 - u ^{2}} 1 - u^{2} + 3 1 - u^{2} = 0 \cdot 1 - u^{2}

化简

u + 3 1 - u^{2} = 0

去除平方根

u + 3 1 - u^{2} = 0

两边减去

u

u + 3 1 - u^{2} - u = 0 - u

化简

3 1 - u^{2} = - u

两边进行平方:

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

u + 3 1 - u^{2} = 0

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

展开

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

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

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

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

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

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

使用指数法则:

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

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

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

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

使用根式运算法则:

a = a^{\frac{1}{2}}

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

使用指数法则:

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

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 1 - u^{2}

= 3 (1 - u^{2})

展开

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

3 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

3 \cdot 1 = 3

= 3 - 3 u^{2}

= 3 - 3 u^{2}

展开

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

(- u)^{2}

使用指数法则:

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

若

n

是偶数

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

= u^{2}

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

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

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

解

3 - 3 u^{2} = u^{2} : u = \frac{3}{2}, u = - \frac{3}{2}

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

将

3

到右边

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

两边减去

3

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

化简

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

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

将

u^{2}

para o lado esquerdo

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

两边减去

u^{2}

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

化简

- 4 u^{2} = - 3

- 4 u^{2} = - 3

两边除以

- 4

- 4 u^{2} = - 3

两边除以

- 4

\frac{- 4 u ^{2}}{- 4} = \frac{- 3}{- 4}

化简

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

化简

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

验证解:

u = \frac{3}{2}

假

, u = - \frac{3}{2}

真

将它们代入

\frac{u}{1 - u ^{2}} + 3 = 0

检验解是否符合

去除与方程不符的解。

代入

u = \frac{3}{2} :

假

\frac{( \frac{3}{2} )}{1 - ( \frac{3}{2} ) ^{2}} + 3 = 0

\frac{( \frac{3}{2} )}{1 - ( \frac{3}{2} ) ^{2}} + 3 = 23

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

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

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

使用分式法则:

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

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

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

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

(\frac{3}{2})^{2} = \frac{3}{4}

(\frac{3}{2})^{2}

使用指数法则:

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

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

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

使用指数法则:

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

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

化简

1 - \frac{3}{4} : \frac{1}{4}

1 - \frac{3}{4}

将项转换为分式:

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

= \frac{1 \cdot 4}{4} - \frac{3}{4}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 4 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

数字相乘:

1 \cdot 4 = 4

= 4 - 3

数字相减:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{1}{2}

使用法则

1 = 1

= \frac{1}{2}

= \frac{3}{2 \cdot \frac{1}{2}}

乘

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= \frac{3}{1}

使用法则

\frac{a}{1} = a

= 3

= 3 + 3

同类项相加:

3 + 3 = 23

= 23

23 = 0

假

代入

u = - \frac{3}{2} :

真

\frac{( - \frac{3}{2} )}{1 - ( - \frac{3}{2} ) ^{2}} + 3 = 0

\frac{( - \frac{3}{2} )}{1 - ( - \frac{3}{2} ) ^{2}} + 3 = 0

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

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

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

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

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

使用指数法则:

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

若

n

是偶数

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

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

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

使用分式法则:

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

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

使用分式法则:

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

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

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

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

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

(\frac{3}{2})^{2} = \frac{3}{4}

(\frac{3}{2})^{2}

使用指数法则:

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

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

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

使用指数法则:

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

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

化简

1 - \frac{3}{4} : \frac{1}{4}

1 - \frac{3}{4}

将项转换为分式:

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

= \frac{1 \cdot 4}{4} - \frac{3}{4}

因为分母相等，所以合并分式:

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

= \frac{1 \cdot 4 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

数字相乘:

1 \cdot 4 = 4

= 4 - 3

数字相减:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{1}{2}

使用法则

1 = 1

= \frac{1}{2}

= - \frac{3}{2 \cdot \frac{1}{2}}

乘

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= - \frac{3}{1}

使用法则

\frac{a}{1} = a

= - 3

= - 3 + 3

同类项相加:

- 3 + 3 = 0

= 0

0 = 0

真

解是

u = - \frac{3}{2}

u = cos (x)

代回

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

合并所有解

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

作图

流行的例子

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