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受欢迎的三角函数 >

sec^2(θ)+5sec(θ)-6=0

解答

sec^{2} (θ) + 5 sec (θ) - 6 = 0

解答

θ = 2 πn, θ = 1.73824 \dots + 2 πn, θ = - 1.73824 \dots + 2 πn

+1

度数

θ = 0^{\circ} + 36 0^{\circ} n, θ = 99.59406 \dots^{\circ} + 36 0^{\circ} n, θ = - 99.59406 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sec^{2} (θ) + 5 sec (θ) - 6 = 0

用替代法求解

sec^{2} (θ) + 5 sec (θ) - 6 = 0

令

: sec (θ) = u

u^{2} + 5 u - 6 = 0

u^{2} + 5 u - 6 = 0 : u = 1, u = - 6

u^{2} + 5 u - 6 = 0

使用求根公式求解

u^{2} + 5 u - 6 = 0

二次方程求根公式:

若

a = 1, b = 5, c = - 6

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

5^{2} - 4 \cdot 1 \cdot (- 6) = 7

5^{2} - 4 \cdot 1 \cdot (- 6)

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 1 \cdot 6

数字相乘:

4 \cdot 1 \cdot 6 = 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:

n a^{n} = a

7^{2} = 7

= 7

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

将解分隔开

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

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

\frac{- 5 + 7}{2 \cdot 1}

数字相加/相减:

- 5 + 7 = 2

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

数字相乘:

2 \cdot 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- 5 - 7}{2 \cdot 1} : - 6

\frac{- 5 - 7}{2 \cdot 1}

数字相减:

- 5 - 7 = - 12

= \frac{- 12}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 12}{2}

使用分式法则:

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

= - \frac{12}{2}

数字相除:

\frac{12}{2} = 6

= - 6

二次方程组的解是:

u = 1, u = - 6

u = sec (θ)

代回

sec (θ) = 1, sec (θ) = - 6

sec (θ) = 1, sec (θ) = - 6

sec (θ) = 1 : θ = 2 πn

sec (θ) = 1

sec (θ) = 1

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}

θ = 0 + 2 πn

θ = 0 + 2 πn

解

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn

sec (θ) = - 6 : θ = arcsec (- 6) + 2 πn, θ = - arcsec (- 6) + 2 πn

sec (θ) = - 6

使用反三角函数性质

sec (θ) = - 6

sec (θ) = - 6

的通解

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

θ = arcsec (- 6) + 2 πn, θ = - arcsec (- 6) + 2 πn

θ = arcsec (- 6) + 2 πn, θ = - arcsec (- 6) + 2 πn

合并所有解

θ = 2 πn, θ = arcsec (- 6) + 2 πn, θ = - arcsec (- 6) + 2 πn

以小数形式表示解

θ = 2 πn, θ = 1.73824 \dots + 2 πn, θ = - 1.73824 \dots + 2 πn

作图

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