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受欢迎的三角函数 >

sec^2(x)-6sec(x)=16

解答

sec^{2} (x) - 6 sec (x) = 16

解答

x = 1.44546 \dots + 2 πn, x = 2 π - 1.44546 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度数

x = 82.81924 \dots^{\circ} + 36 0^{\circ} n, x = 277.18075 \dots^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

sec^{2} (x) - 6 sec (x) = 16

用替代法求解

sec^{2} (x) - 6 sec (x) = 16

令

: sec (x) = u

u^{2} - 6 u = 16

u^{2} - 6 u = 16 : u = 8, u = - 2

u^{2} - 6 u = 16

将

16

para o lado esquerdo

u^{2} - 6 u = 16

两边减去

16

u^{2} - 6 u - 16 = 16 - 16

化简

u^{2} - 6 u - 16 = 0

u^{2} - 6 u - 16 = 0

使用求根公式求解

u^{2} - 6 u - 16 = 0

二次方程求根公式:

若

a = 1, b = - 6, c = - 16

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot ( - 16 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot ( - 16 )}{2 \cdot 1}

(- 6)^{2} - 4 \cdot 1 \cdot (- 16) = 10

(- 6)^{2} - 4 \cdot 1 \cdot (- 16)

使用法则

- (- a) = a

= (- 6)^{2} + 4 \cdot 1 \cdot 16

使用指数法则:

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

若

n

是偶数

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 1 \cdot 16

数字相乘:

4 \cdot 1 \cdot 16 = 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 1}

将解分隔开

u_{1} = \frac{- ( - 6 ) + 10}{2 \cdot 1}, u_{2} = \frac{- ( - 6 ) - 10}{2 \cdot 1}

u = \frac{- ( - 6 ) + 10}{2 \cdot 1} : 8

\frac{- ( - 6 ) + 10}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{6 + 10}{2 \cdot 1}

数字相加:

6 + 10 = 16

= \frac{16}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{16}{2}

数字相除:

\frac{16}{2} = 8

= 8

u = \frac{- ( - 6 ) - 10}{2 \cdot 1} : - 2

\frac{- ( - 6 ) - 10}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{6 - 10}{2 \cdot 1}

数字相减:

6 - 10 = - 4

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 4}{2}

使用分式法则:

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

= - \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= - 2

二次方程组的解是:

u = 8, u = - 2

u = sec (x)

代回

sec (x) = 8, sec (x) = - 2

sec (x) = 8, sec (x) = - 2

sec (x) = 8 : x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn

sec (x) = 8

使用反三角函数性质

sec (x) = 8

sec (x) = 8

的通解

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

x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn

x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn

sec (x) = - 2 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

sec (x) = - 2

sec (x) = - 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

合并所有解

x = arcsec (8) + 2 πn, x = 2 π - arcsec (8) + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

以小数形式表示解

x = 1.44546 \dots + 2 πn, x = 2 π - 1.44546 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

作图

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