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受欢迎的三角函数 >

3tan^2(x)-sec(x)-1=0

解答

3 tan^{2} (x) - sec (x) - 1 = 0

解答

x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn, x = π + 2 πn

+1

度数

x = 41.40962 \dots^{\circ} + 36 0^{\circ} n, x = 318.59037 \dots^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

3 tan^{2} (x) - sec (x) - 1 = 0

使用三角恒等式改写

- 1 - sec (x) + 3 tan^{2} (x)

使用毕达哥拉斯恒等式:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= - 1 - sec (x) + 3 (sec^{2} (x) - 1)

化简

- 1 - sec (x) + 3 (sec^{2} (x) - 1) : 3 sec^{2} (x) - sec (x) - 4

- 1 - sec (x) + 3 (sec^{2} (x) - 1)

乘开

3 (sec^{2} (x) - 1) : 3 sec^{2} (x) - 3

3 (sec^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = 3, b = sec^{2} (x), c = 1

= 3 sec^{2} (x) - 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

= 3 sec^{2} (x) - 3

= - 1 - sec (x) + 3 sec^{2} (x) - 3

化简

- 1 - sec (x) + 3 sec^{2} (x) - 3 : 3 sec^{2} (x) - sec (x) - 4

- 1 - sec (x) + 3 sec^{2} (x) - 3

对同类项分组

= - sec (x) + 3 sec^{2} (x) - 1 - 3

数字相减:

- 1 - 3 = - 4

= 3 sec^{2} (x) - sec (x) - 4

= 3 sec^{2} (x) - sec (x) - 4

= 3 sec^{2} (x) - sec (x) - 4

- 4 - sec (x) + 3 sec^{2} (x) = 0

用替代法求解

- 4 - sec (x) + 3 sec^{2} (x) = 0

令

: sec (x) = u

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

- 4 - u + 3 u^{2} = 0 : u = \frac{4}{3}, u = - 1

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

改写成标准形式

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

3 u^{2} - u - 4 = 0

使用求根公式求解

3 u^{2} - u - 4 = 0

二次方程求根公式:

若

a = 3, b = - 1, c = - 4

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

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

(- 1)^{2} - 4 \cdot 3 (- 4) = 7

(- 1)^{2} - 4 \cdot 3 (- 4)

使用法则

- (- a) = a

= (- 1)^{2} + 4 \cdot 3 \cdot 4

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 3 \cdot 4 = 48

4 \cdot 3 \cdot 4

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 48

= 1 + 48

数字相加:

1 + 48 = 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{- ( - 1 ) \pm 7}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 7}{2 \cdot 3}, u_{2} = \frac{- ( - 1 ) - 7}{2 \cdot 3}

u = \frac{- ( - 1 ) + 7}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 1 ) + 7}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{1 + 7}{2 \cdot 3}

数字相加:

1 + 7 = 8

= \frac{8}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{8}{6}

约分:

2

= \frac{4}{3}

u = \frac{- ( - 1 ) - 7}{2 \cdot 3} : - 1

\frac{- ( - 1 ) - 7}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{1 - 7}{2 \cdot 3}

数字相减:

1 - 7 = - 6

= \frac{- 6}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 6}{6}

使用分式法则:

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

= - \frac{6}{6}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = \frac{4}{3}, u = - 1

u = sec (x)

代回

sec (x) = \frac{4}{3}, sec (x) = - 1

sec (x) = \frac{4}{3}, sec (x) = - 1

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

sec (x) = \frac{4}{3}

使用反三角函数性质

sec (x) = \frac{4}{3}

sec (x) = \frac{4}{3}

的通解

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

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

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

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

sec (x) = - 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}

x = π + 2 πn

x = π + 2 πn

合并所有解

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

以小数形式表示解

x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn, x = π + 2 πn

作图

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