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受欢迎的三角函数 >

sec^2(x)-7sec(x)=0

解答

sec^{2} (x) - 7 sec (x) = 0

解答

x = 1.42744 \dots + 2 πn, x = 2 π - 1.42744 \dots + 2 πn

+1

度数

x = 81.78678 \dots^{\circ} + 36 0^{\circ} n, x = 278.21321 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sec^{2} (x) - 7 sec (x) = 0

用替代法求解

sec^{2} (x) - 7 sec (x) = 0

令

: sec (x) = u

u^{2} - 7 u = 0

u^{2} - 7 u = 0 : u = 7, u = 0

u^{2} - 7 u = 0

使用求根公式求解

u^{2} - 7 u = 0

二次方程求根公式:

若

a = 1, b = - 7, c = 0

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 7)^{2} - 4 \cdot 1 \cdot 0 = 7

(- 7)^{2} - 4 \cdot 1 \cdot 0

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 1 \cdot 0

使用法则

0 \cdot a = 0

= 7^{2} - 0

7^{2} - 0 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 7

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相加:

7 + 7 = 14

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

数字相乘:

2 \cdot 1 = 2

= \frac{14}{2}

数字相除:

\frac{14}{2} = 7

= 7

u = \frac{- ( - 7 ) - 7}{2 \cdot 1} : 0

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

使用法则

- (- a) = a

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

数字相减:

7 - 7 = 0

= \frac{0}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{0}{2}

使用法则

\frac{0}{a} = 0, a \neq = 0

= 0

二次方程组的解是:

u = 7, u = 0

u = sec (x)

代回

sec (x) = 7, sec (x) = 0

sec (x) = 7, sec (x) = 0

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

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

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

sec (x) = 0 :

无解

sec (x) = 0

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

无解

合并所有解

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

以小数形式表示解

x = 1.42744 \dots + 2 πn, x = 2 π - 1.42744 \dots + 2 πn

作图

流行的例子

3 cot (θ) - 7 = 0

cos(x)= 1249/1250

cos (x) = \frac{1249}{1250}

sec (θ) = \frac{9}{7}

4 cos (x - π) = 4

cot(x)(tan(x)-(sqrt(3))/3)=0

cot (x) (tan (x) - \frac{3}{3}) = 0