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cos^2(x)+sec^2(x)=11

解答

cos^{2} (x) + sec^{2} (x) = 11

解答

x = 1.26319 \dots + 2 πn, x = 2 π - 1.26319 \dots + 2 πn, x = 1.87839 \dots + 2 πn, x = - 1.87839 \dots + 2 πn

+1

度数

x = 72.37560 \dots^{\circ} + 36 0^{\circ} n, x = 287.62439 \dots^{\circ} + 36 0^{\circ} n, x = 107.62439 \dots^{\circ} + 36 0^{\circ} n, x = - 107.62439 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) + sec^{2} (x) = 11

两边减去

11

cos^{2} (x) + sec^{2} (x) - 11 = 0

使用三角恒等式改写

- 11 + cos^{2} (x) + sec^{2} (x)

使用基本三角恒等式:

cos (x) = \frac{1}{sec ( x )}

= - 11 + (\frac{1}{sec ( x )})^{2} + sec^{2} (x)

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{sec ^{2} ( x )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

= - 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x)

- 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x) = 0

用替代法求解

- 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x) = 0

令

: sec (x) = u

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

- 11 + \frac{1}{u ^{2}} + u^{2} = 0 : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

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

在两边乘以

u^{2}

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

在两边乘以

u^{2}

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

化简

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

化简

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

分式相乘:

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

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

约分:

u^{2}

= 1

化简

u^{2} u^{2} : u^{4}

u^{2} u^{2}

使用指数法则:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

数字相加:

2 + 2 = 4

= u^{4}

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

- 11 u^{2} + 1 + u^{4} = 0

- 11 u^{2} + 1 + u^{4} = 0

- 11 u^{2} + 1 + u^{4} = 0

解

- 11 u^{2} + 1 + u^{4} = 0 : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

- 11 u^{2} + 1 + u^{4} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - 11 u^{2} + 1 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} - 11 v + 1 = 0

解

v^{2} - 11 v + 1 = 0 : v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

v^{2} - 11 v + 1 = 0

使用求根公式求解

v^{2} - 11 v + 1 = 0

二次方程求根公式:

若

a = 1, b = - 11, c = 1

v_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

(- 11)^{2} - 4 \cdot 1 \cdot 1 = 313

(- 11)^{2} - 4 \cdot 1 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 11)^{2} = 1 1^{2}

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

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 1 1^{2} - 4

1 1^{2} = 121

= 121 - 4

数字相减:

121 - 4 = 117

= 117

117

质因数分解:

3^{2} \cdot 13

117

117

除以

3117 = 39 \cdot 3

= 3 \cdot 39

39

除以

339 = 13 \cdot 3

= 3 \cdot 3 \cdot 13

3, 13

都是质数，因此无法进一步因数分解

= 3 \cdot 3 \cdot 13

= 3^{2} \cdot 13

= 3^{2} \cdot 13

使用根式运算法则:

= 13 3^{2}

使用根式运算法则:

3^{2} = 3

= 313

v_{1, 2} = \frac{- ( - 11 ) \pm 3 13}{2 \cdot 1}

将解分隔开

v_{1} = \frac{- ( - 11 ) + 3 13}{2 \cdot 1}, v_{2} = \frac{- ( - 11 ) - 3 13}{2 \cdot 1}

v = \frac{- ( - 11 ) + 3 13}{2 \cdot 1} : \frac{11 + 3 13}{2}

\frac{- ( - 11 ) + 3 13}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{11 + 3 13}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{11 + 3 13}{2}

v = \frac{- ( - 11 ) - 3 13}{2 \cdot 1} : \frac{11 - 3 13}{2}

\frac{- ( - 11 ) - 3 13}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{11 - 3 13}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{11 - 3 13}{2}

二次方程组的解是:

v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

代回

v = u^{2}

，求解

u

解

u^{2} = \frac{11 + 3 13}{2} : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}

u^{2} = \frac{11 + 3 13}{2}

对于

x^{2} = f (a)

解为

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

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}

解

u^{2} = \frac{11 - 3 13}{2} : u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

u^{2} = \frac{11 - 3 13}{2}

对于

x^{2} = f (a)

解为

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

u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

解为

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

验证解

找到无定义的点（奇点）:

u = 0

取

- 11 + \frac{1}{u ^{2}} + u^{2}

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

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{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

u = sec (x)

代回

sec (x) = \frac{11 + 3 13}{2}, sec (x) = - \frac{11 + 3 13}{2}, sec (x) = \frac{11 - 3 13}{2}, sec (x) = - \frac{11 - 3 13}{2}

sec (x) = \frac{11 + 3 13}{2}, sec (x) = - \frac{11 + 3 13}{2}, sec (x) = \frac{11 - 3 13}{2}, sec (x) = - \frac{11 - 3 13}{2}

sec (x) = \frac{11 + 3 13}{2} : x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

sec (x) = \frac{11 + 3 13}{2}

使用反三角函数性质

sec (x) = \frac{11 + 3 13}{2}

sec (x) = \frac{11 + 3 13}{2}

的通解

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

sec (x) = - \frac{11 + 3 13}{2} : x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

sec (x) = - \frac{11 + 3 13}{2}

使用反三角函数性质

sec (x) = - \frac{11 + 3 13}{2}

sec (x) = - \frac{11 + 3 13}{2}

的通解

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

sec (x) = \frac{11 - 3 13}{2} :

无解

sec (x) = \frac{11 - 3 13}{2}

sec (x) \leq - 1 or sec (x) \geq 1

无解

sec (x) = - \frac{11 - 3 13}{2} :

无解

sec (x) = - \frac{11 - 3 13}{2}

sec (x) \leq - 1 or sec (x) \geq 1

无解

合并所有解

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn, x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

以小数形式表示解

x = 1.26319 \dots + 2 πn, x = 2 π - 1.26319 \dots + 2 πn, x = 1.87839 \dots + 2 πn, x = - 1.87839 \dots + 2 πn

流行的例子

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csc (2 x + 3^{\circ}) = \frac{5}{4}

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-sqrt(2)sin(2x)=sin^2(x)+cos^2(x)

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