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受欢迎的三角函数 >

4/(cos(x))-6=tan(x)

解答

\frac{4}{cos ( x )} - 6 = tan (x)

解答

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

+1

度数

x = 320.57910 \dots^{\circ} + 36 0^{\circ} n, x = 58.34553 \dots^{\circ} + 36 0^{\circ} n

求解步骤

\frac{4}{cos ( x )} - 6 = tan (x)

两边减去

tan (x)

\frac{4}{cos ( x )} - 6 - tan (x) = 0

化简

\frac{4}{cos ( x )} - 6 - tan (x) : \frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4}{cos ( x )} - 6 - tan (x)

将项转换为分式:

6 = \frac{6 cos ( x )}{cos ( x )}, tan (x) = \frac{tan ( x ) cos ( x )}{cos ( x )}

= \frac{4}{cos ( x )} - \frac{6 cos ( x )}{cos ( x )} - \frac{tan ( x ) cos ( x )}{cos ( x )}

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{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

4 - 6 cos (x) - tan (x) cos (x) = 0

用 sin, cos 表示

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) = 0

化简

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) : 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x)

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

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

分式相乘:

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

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

约分:

cos (x)

= sin (x)

= 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - sin (x) = 0

两边加上

sin (x)

4 - 6 cos (x) = sin (x)

两边进行平方

(4 - 6 cos (x))^{2} = sin^{2} (x)

两边减去

sin^{2} (x)

(4 - 6 cos (x))^{2} - sin^{2} (x) = 0

使用三角恒等式改写

(4 - 6 cos (x))^{2} - sin^{2} (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

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

= (4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

化简

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x)) : 37 cos^{2} (x) - 48 cos (x) + 15

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

(4 - 6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 4, b = 6 cos (x)

= 4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

化简

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

2 \cdot 4 \cdot 6 cos (x) = 48 cos (x)

2 \cdot 4 \cdot 6 cos (x)

数字相乘:

2 \cdot 4 \cdot 6 = 48

= 48 cos (x)

(6 cos (x))^{2} = 36 cos^{2} (x)

(6 cos (x))^{2}

使用指数法则:

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

= 6^{2} cos^{2} (x)

6^{2} = 36

= 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x) - (1 - cos^{2} (x))

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

= 16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

化简

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x) : 37 cos^{2} (x) - 48 cos (x) + 15

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

对同类项分组

= - 48 cos (x) + 36 cos^{2} (x) + cos^{2} (x) + 16 - 1

同类项相加:

36 cos^{2} (x) + cos^{2} (x) = 37 cos^{2} (x)

= - 48 cos (x) + 37 cos^{2} (x) + 16 - 1

数字相加/相减:

16 - 1 = 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

15 + 37 cos^{2} (x) - 48 cos (x) = 0

用替代法求解

15 + 37 cos^{2} (x) - 48 cos (x) = 0

令

: cos (x) = u

15 + 37 u^{2} - 48 u = 0

15 + 37 u^{2} - 48 u = 0 : u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

15 + 37 u^{2} - 48 u = 0

改写成标准形式

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

37 u^{2} - 48 u + 15 = 0

使用求根公式求解

37 u^{2} - 48 u + 15 = 0

二次方程求根公式:

若

a = 37, b = - 48, c = 15

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

(- 48)^{2} - 4 \cdot 37 \cdot 15 = 221

(- 48)^{2} - 4 \cdot 37 \cdot 15

使用指数法则:

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

若

n

是偶数

(- 48)^{2} = 4 8^{2}

= 4 8^{2} - 4 \cdot 37 \cdot 15

数字相乘:

4 \cdot 37 \cdot 15 = 2220

= 4 8^{2} - 2220

4 8^{2} = 2304

= 2304 - 2220

数字相减:

2304 - 2220 = 84

= 84

84

质因数分解:

2^{2} \cdot 3 \cdot 7

84

84

除以

284 = 42 \cdot 2

= 2 \cdot 42

42

除以

242 = 21 \cdot 2

= 2 \cdot 2 \cdot 21

21

除以

321 = 7 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 7

2, 3, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 7

= 2^{2} \cdot 3 \cdot 7

= 2^{2} \cdot 3 \cdot 7

使用根式运算法则:

n ab = n a n b

= 2^{2} 3 \cdot 7

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 3 \cdot 7

整理后得

= 221

u_{1, 2} = \frac{- ( - 48 ) \pm 2 21}{2 \cdot 37}

将解分隔开

u_{1} = \frac{- ( - 48 ) + 2 21}{2 \cdot 37}, u_{2} = \frac{- ( - 48 ) - 2 21}{2 \cdot 37}

u = \frac{- ( - 48 ) + 2 21}{2 \cdot 37} : \frac{24 + 21}{37}

\frac{- ( - 48 ) + 2 21}{2 \cdot 37}

使用法则

- (- a) = a

= \frac{48 + 2 21}{2 \cdot 37}

数字相乘:

2 \cdot 37 = 74

= \frac{48 + 2 21}{74}

分解

48 + 221 : 2 (24 + 21)

48 + 221

改写为

= 2 \cdot 24 + 221

因式分解出通项

2

= 2 (24 + 21)

= \frac{2 ( 24 + 21 )}{74}

约分:

2

= \frac{24 + 21}{37}

u = \frac{- ( - 48 ) - 2 21}{2 \cdot 37} : \frac{24 - 21}{37}

\frac{- ( - 48 ) - 2 21}{2 \cdot 37}

使用法则

- (- a) = a

= \frac{48 - 2 21}{2 \cdot 37}

数字相乘:

2 \cdot 37 = 74

= \frac{48 - 2 21}{74}

分解

48 - 221 : 2 (24 - 21)

48 - 221

改写为

= 2 \cdot 24 - 221

因式分解出通项

2

= 2 (24 - 21)

= \frac{2 ( 24 - 21 )}{74}

约分:

2

= \frac{24 - 21}{37}

二次方程组的解是:

u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

u = cos (x)

代回

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37} : x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 + 21}{37}

使用反三角函数性质

cos (x) = \frac{24 + 21}{37}

cos (x) = \frac{24 + 21}{37}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37} : x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37}

使用反三角函数性质

cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 - 21}{37}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

合并所有解

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

将解代入原方程进行验证

将它们代入

\frac{4}{cos ( x )} - 6 = tan (x)

检验解是否符合

去除与方程不符的解。

检验

arccos (\frac{24 + 21}{37}) + 2 πn

的解:

假

arccos (\frac{24 + 21}{37}) + 2 πn

代入

n = 1

arccos (\frac{24 + 21}{37}) + 2 π 1

对于

\frac{4}{cos ( x )} - 6 = tan (x)

代入

x = arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 + 21}{37}) + 2 π 1)

整理后得

- 0.82202 \dots = 0.82202 \dots

\Rightarrow 假

检验

2 π - arccos (\frac{24 + 21}{37}) + 2 πn

的解:

真

2 π - arccos (\frac{24 + 21}{37}) + 2 πn

代入

n = 1

2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

对于

\frac{4}{cos ( x )} - 6 = tan (x)

代入

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 + 21}{37}) + 2 π 1)

整理后得

- 0.82202 \dots = - 0.82202 \dots

\Rightarrow 真

检验

arccos (\frac{24 - 21}{37}) + 2 πn

的解:

真

arccos (\frac{24 - 21}{37}) + 2 πn

代入

n = 1

arccos (\frac{24 - 21}{37}) + 2 π 1

对于

\frac{4}{cos ( x )} - 6 = tan (x)

代入

x = arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 - 21}{37}) + 2 π 1)

整理后得

1.62202 \dots = 1.62202 \dots

\Rightarrow 真

检验

2 π - arccos (\frac{24 - 21}{37}) + 2 πn

的解:

假

2 π - arccos (\frac{24 - 21}{37}) + 2 πn

代入

n = 1

2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

对于

\frac{4}{cos ( x )} - 6 = tan (x)

代入

x = 2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 - 21}{37}) + 2 π 1)

整理后得

1.62202 \dots = - 1.62202 \dots

\Rightarrow 假

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn

以小数形式表示解

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

作图

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