zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

2sec^2(x)+3(1/(cos(x)))=2

解答

2 sec^{2} (x) + 3 (\frac{1}{cos ( x )}) = 2

解答

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

2 sec^{2} (x) + 3 (\frac{1}{cos ( x )}) = 2

两边减去

2

2 sec^{2} (x) + \frac{3}{cos ( x )} - 2 = 0

化简

2 sec^{2} (x) + \frac{3}{cos ( x )} - 2 : \frac{2 sec ^{2} ( x ) cos ( x ) + 3 - 2 cos ( x )}{cos ( x )}

2 sec^{2} (x) + \frac{3}{cos ( x )} - 2

将项转换为分式:

2 sec^{2} (x) = \frac{2 sec ^{2} ( x ) cos ( x )}{cos ( x )}, 2 = \frac{2 cos ( x )}{cos ( x )}

= \frac{2 sec ^{2} ( x ) cos ( x )}{cos ( x )} + \frac{3}{cos ( x )} - \frac{2 cos ( x )}{cos ( x )}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 sec ^{2} ( x ) cos ( x ) + 3 - 2 cos ( x )}{cos ( x )}

\frac{2 sec ^{2} ( x ) cos ( x ) + 3 - 2 cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

使用三角恒等式改写

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

使用基本三角恒等式:

cos (x) = \frac{1}{sec ( x )}

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

化简

3 - 2 \cdot \frac{1}{sec ( x )} + 2 \cdot \frac{1}{sec ( x )} sec^{2} (x) : 3 - \frac{2}{sec ( x )} + 2 sec (x)

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

2 \cdot \frac{1}{sec ( x )} = \frac{2}{sec ( x )}

2 \cdot \frac{1}{sec ( x )}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{sec ( x )}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{sec ( x )}

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

2 \cdot \frac{1}{sec ( x )} sec^{2} (x)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

数字相乘:

1 \cdot 2 = 2

= \frac{2 sec ^{2} ( x )}{sec ( x )}

约分:

sec (x)

= 2 sec (x)

= 3 - \frac{2}{sec ( x )} + 2 sec (x)

= 3 - \frac{2}{sec ( x )} + 2 sec (x)

3 - \frac{2}{sec ( x )} + 2 sec (x) = 0

用替代法求解

3 - \frac{2}{sec ( x )} + 2 sec (x) = 0

令

: sec (x) = u

3 - \frac{2}{u} + 2 u = 0

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

3 - \frac{2}{u} + 2 u = 0

在两边乘以

u

3 - \frac{2}{u} + 2 u = 0

在两边乘以

u

3 u - \frac{2}{u} u + 2 uu = 0 \cdot u

化简

3 u - \frac{2}{u} u + 2 uu = 0 \cdot u

化简

- \frac{2}{u} u : - 2

- \frac{2}{u} u

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{2 u}{u}

约分:

u

= - 2

化简

2 uu : 2 u^{2}

2 uu

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 2 u^{1 + 1}

数字相加:

1 + 1 = 2

= 2 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

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

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

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

解

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 2, b = 3, c = - 2

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

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

3^{2} - 4 \cdot 2 (- 2) = 5

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

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 2 \cdot 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 3 + 5}{2 \cdot 2}, u_{2} = \frac{- 3 - 5}{2 \cdot 2}

u = \frac{- 3 + 5}{2 \cdot 2} : \frac{1}{2}

\frac{- 3 + 5}{2 \cdot 2}

数字相加/相减:

- 3 + 5 = 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

约分:

2

= \frac{1}{2}

u = \frac{- 3 - 5}{2 \cdot 2} : - 2

\frac{- 3 - 5}{2 \cdot 2}

数字相减:

- 3 - 5 = - 8

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 8}{4}

使用分式法则:

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

= - \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= - 2

二次方程组的解是:

u = \frac{1}{2}, u = - 2

u = \frac{1}{2}, u = - 2

验证解

找到无定义的点（奇点）:

u = 0

取

3 - \frac{2}{u} + 2 u

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{1}{2}, u = - 2

u = sec (x)

代回

sec (x) = \frac{1}{2}, sec (x) = - 2

sec (x) = \frac{1}{2}, sec (x) = - 2

sec (x) = \frac{1}{2} :

无解

sec (x) = \frac{1}{2}

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

无解

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

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

合并所有解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

作图

流行的例子

cot^{2} (θ) - 3 = 0

tan(θ)=2,sin(θ)

tan (θ) = 2, sin (θ)

7sin^2(x)+14sin(x)+7=0

7 sin^{2} (x) + 14 sin (x) + 7 = 0

tan (θ) = 17

tan (θ) = 14