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受欢迎的三角函数 >

sec^2(x)+5sec(x)-14=0

解答

sec^{2} (x) + 5 sec (x) - 14 = 0

解答

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 1.71414 \dots + 2 πn, x = - 1.71414 \dots + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 98.21321 \dots^{\circ} + 36 0^{\circ} n, x = - 98.21321 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sec^{2} (x) + 5 sec (x) - 14 = 0

用替代法求解

sec^{2} (x) + 5 sec (x) - 14 = 0

令

: sec (x) = u

u^{2} + 5 u - 14 = 0

u^{2} + 5 u - 14 = 0 : u = 2, u = - 7

u^{2} + 5 u - 14 = 0

使用求根公式求解

u^{2} + 5 u - 14 = 0

二次方程求根公式:

若

a = 1, b = 5, c = - 14

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot ( - 14 )}{2 \cdot 1}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot ( - 14 )}{2 \cdot 1}

5^{2} - 4 \cdot 1 \cdot (- 14) = 9

5^{2} - 4 \cdot 1 \cdot (- 14)

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 1 \cdot 14

数字相乘:

4 \cdot 1 \cdot 14 = 56

= 5^{2} + 56

5^{2} = 25

= 25 + 56

数字相加:

25 + 56 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 5 \pm 9}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 5 + 9}{2 \cdot 1}, u_{2} = \frac{- 5 - 9}{2 \cdot 1}

u = \frac{- 5 + 9}{2 \cdot 1} : 2

\frac{- 5 + 9}{2 \cdot 1}

数字相加/相减:

- 5 + 9 = 4

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

数字相乘:

2 \cdot 1 = 2

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

u = \frac{- 5 - 9}{2 \cdot 1} : - 7

\frac{- 5 - 9}{2 \cdot 1}

数字相减:

- 5 - 9 = - 14

= \frac{- 14}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 14}{2}

使用分式法则:

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

= - \frac{14}{2}

数字相除:

\frac{14}{2} = 7

= - 7

二次方程组的解是:

u = 2, u = - 7

u = sec (x)

代回

sec (x) = 2, sec (x) = - 7

sec (x) = 2, sec (x) = - 7

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = - 7 : x = arcsec (- 7) + 2 πn, x = - arcsec (- 7) + 2 πn

sec (x) = - 7

使用反三角函数性质

sec (x) = - 7

sec (x) = - 7

的通解

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

x = arcsec (- 7) + 2 πn, x = - arcsec (- 7) + 2 πn

x = arcsec (- 7) + 2 πn, x = - arcsec (- 7) + 2 πn

合并所有解

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

以小数形式表示解

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 1.71414 \dots + 2 πn, x = - 1.71414 \dots + 2 πn

作图

流行的例子

3 cos^{2} (x) = 1

cos(θ)=-7/8 ,csc(θ)

cos (θ) = - \frac{7}{8}, csc (θ)

1+2sin(x)-cos^2(x)=0

1 + 2 sin (x) - cos^{2} (x) = 0

tan(θ)=-3,sin(θ)<0

tan (θ) = - 3, sin (θ) < 0

solvefor x,sec(x)+sqrt(2)=0

so l v e f or x, sec (x) + 2 = 0