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受欢迎的三角函数 >

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

解答

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

解答

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

+1

度数

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sec (x) = u

u^{2} - 3 u = 0

u^{2} - 3 u = 0 : u = 3, u = 0

u^{2} - 3 u = 0

使用求根公式求解

u^{2} - 3 u = 0

二次方程求根公式:

若

a = 1, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 3)^{2} - 4 \cdot 1 \cdot 0 = 3

(- 3)^{2} - 4 \cdot 1 \cdot 0

使用指数法则:

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

若

n

是偶数

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

= 3^{2} - 4 \cdot 1 \cdot 0

使用法则

0 \cdot a = 0

= 3^{2} - 0

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

= 3^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 3

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相加:

3 + 3 = 6

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

数字相乘:

2 \cdot 1 = 2

= \frac{6}{2}

数字相除:

\frac{6}{2} = 3

= 3

u = \frac{- ( - 3 ) - 3}{2 \cdot 1} : 0

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

使用法则

- (- a) = a

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

数字相减:

3 - 3 = 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 = 3, u = 0

u = sec (x)

代回

sec (x) = 3, sec (x) = 0

sec (x) = 3, sec (x) = 0

sec (x) = 3 : x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) = 3

使用反三角函数性质

sec (x) = 3

sec (x) = 3

的通解

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

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 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 (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

以小数形式表示解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

作图

流行的例子

sqrt(3)cot(x)+1=0

3 cot (x) + 1 = 0

tan (x) = - \frac{2}{3}

tan (θ) = \frac{5}{3}

tan (x) = \frac{3}{5}

6sec^2(θ)tan(θ)=12tan(θ)

6 sec^{2} (θ) tan (θ) = 12 tan (θ)