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3cos(x)=2sec(x)-5

解答

3 cos (x) = 2 sec (x) - 5

解答

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

+1

度数

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 cos (x) = 2 sec (x) - 5

两边减去

2 sec (x) - 5

3 cos (x) - 2 sec (x) + 5 = 0

使用三角恒等式改写

5 - 2 sec (x) + 3 cos (x)

使用基本三角恒等式:

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

= 5 - 2 sec (x) + 3 \cdot \frac{1}{sec ( x )}

3 \cdot \frac{1}{sec ( x )} = \frac{3}{sec ( x )}

3 \cdot \frac{1}{sec ( x )}

分式相乘:

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

= \frac{1 \cdot 3}{sec ( x )}

数字相乘:

1 \cdot 3 = 3

= \frac{3}{sec ( x )}

= 5 - 2 sec (x) + \frac{3}{sec ( x )}

5 + \frac{3}{sec ( x )} - 2 sec (x) = 0

用替代法求解

5 + \frac{3}{sec ( x )} - 2 sec (x) = 0

令

: sec (x) = u

5 + \frac{3}{u} - 2 u = 0

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

5 + \frac{3}{u} - 2 u = 0

在两边乘以

u

5 + \frac{3}{u} - 2 u = 0

在两边乘以

u

5 u + \frac{3}{u} u - 2 uu = 0 \cdot u

化简

5 u + \frac{3}{u} u - 2 uu = 0 \cdot u

化简

\frac{3}{u} u : 3

\frac{3}{u} u

分式相乘:

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

= \frac{3 u}{u}

约分:

u

= 3

化简

- 2 uu : - 2 u^{2}

- 2 uu

使用指数法则:

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

uu = u^{1 + 1}

= - 2 u^{1 + 1}

数字相加:

1 + 1 = 2

= - 2 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

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

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

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

解

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

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

改写成标准形式

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

- 2 u^{2} + 5 u + 3 = 0

使用求根公式求解

- 2 u^{2} + 5 u + 3 = 0

二次方程求根公式:

若

a = - 2, b = 5, c = 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

5^{2} - 4 (- 2) \cdot 3 = 7

5^{2} - 4 (- 2) \cdot 3

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 2 \cdot 3

数字相乘:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数字相加:

25 + 24 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- 5 + 7}{2 ( - 2 )}, u_{2} = \frac{- 5 - 7}{2 ( - 2 )}

u = \frac{- 5 + 7}{2 ( - 2 )} : - \frac{1}{2}

\frac{- 5 + 7}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 5 + 7}{- 2 \cdot 2}

数字相加/相减:

- 5 + 7 = 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{2}{- 4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

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

\frac{- 5 - 7}{2 ( - 2 )}

去除括号:

(- a) = - a

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

数字相减:

- 5 - 7 = - 12

= \frac{- 12}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 12}{- 4}

使用分式法则:

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

= \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= 3

二次方程组的解是:

u = - \frac{1}{2}, u = 3

u = - \frac{1}{2}, u = 3

验证解

找到无定义的点（奇点）:

u = 0

取

5 + \frac{3}{u} - 2 u

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 = 3

u = sec (x)

代回

sec (x) = - \frac{1}{2}, sec (x) = 3

sec (x) = - \frac{1}{2}, sec (x) = 3

sec (x) = - \frac{1}{2} :

无解

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

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

无解

sec (x) = 3 : x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) = 3

使用反三角函数性质

sec (x) = 3

sec (x) = 3

的通解

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

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

合并所有解

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

以小数形式表示解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

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