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 >

(cot(x)+1/(cot(x)))/(cos(x))=sec(x)

解答

\frac{cot ( x ) + \frac{1}{c o t ( x )}}{cos ( x )} = sec (x)

解答

x \in R 无解

求解步骤

\frac{cot ( x ) + \frac{1}{c o t ( x )}}{cos ( x )} = sec (x)

两边减去

sec (x)

\frac{cot ^{2} ( x ) + 1}{cot ( x ) cos ( x )} - sec (x) = 0

化简

\frac{cot ^{2} ( x ) + 1}{cot ( x ) cos ( x )} - sec (x) : \frac{cot ^{2} ( x ) + 1 - sec ( x ) cot ( x ) cos ( x )}{cot ( x ) cos ( x )}

\frac{cot ^{2} ( x ) + 1}{cot ( x ) cos ( x )} - sec (x)

将项转换为分式:

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

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

\frac{cot ^{2} ( x ) + 1 - sec ( x ) cot ( x ) cos ( x )}{cot ( x ) cos ( x )} = 0

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

cot^{2} (x) + 1 - sec (x) cot (x) cos (x) = 0

使用三角恒等式改写

1 + cot^{2} (x) - cos (x) cot (x) sec (x)

使用基本三角恒等式:

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

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

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

\frac{1}{sec ( x )} cot (x) sec (x)

分式相乘:

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

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

约分:

sec (x)

= 1 \cdot cot (x)

乘以:

1 \cdot cot (x) = cot (x)

= cot (x)

= 1 + cot^{2} (x) - cot (x)

1 - cot (x) + cot^{2} (x) = 0

用替代法求解

1 - cot (x) + cot^{2} (x) = 0

令

: cot (x) = u

1 - u + u^{2} = 0

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

1 - u + u^{2} = 0

改写成标准形式

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

u^{2} - u + 1 = 0

使用求根公式求解

u^{2} - u + 1 = 0

二次方程求根公式:

若

a = 1, b = - 1, c = 1

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

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

化简

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

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

(- 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 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 4

= 1 - 4

数字相减:

1 - 4 = - 3

= - 3

使用根式运算法则:

- a = - 1 a

- 3 = - 1 3

= - 1 3

使用虚数运算法则:

- 1 = i

= 3 i

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

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

将

\frac{1 + 3 i}{2}

改写成标准复数形式:

\frac{1}{2} + \frac{3}{2} i

\frac{1 + 3 i}{2}

使用分式法则:

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

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

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

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

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

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

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

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

将

\frac{1 - 3 i}{2}

改写成标准复数形式:

\frac{1}{2} - \frac{3}{2} i

\frac{1 - 3 i}{2}

使用分式法则:

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

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

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

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

二次方程组的解是:

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

u = cot (x)

代回

cot (x) = \frac{1}{2} + i \frac{3}{2}, cot (x) = \frac{1}{2} - i \frac{3}{2}

cot (x) = \frac{1}{2} + i \frac{3}{2}, cot (x) = \frac{1}{2} - i \frac{3}{2}

cot (x) = \frac{1}{2} + i \frac{3}{2} :

无解

cot (x) = \frac{1}{2} + i \frac{3}{2}

无解

cot (x) = \frac{1}{2} - i \frac{3}{2} :

无解

cot (x) = \frac{1}{2} - i \frac{3}{2}

无解

合并所有解

x \in R 无解

作图

流行的例子

sec(θ)=2,(tan(θ)-sin(θ))^2+(1-cos(θ))^2

sec (θ) = 2, (tan (θ) - sin (θ))^{2} + (1 - cos (θ))^{2}

(1+csc(A))(1-csc(A))=-cot(A)

(1 + csc (A)) (1 - csc (A)) = - cot (A)

6cos^2(3x)-cos(3x)-2=0

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

cos(2x)-3cos(x)+1=0

cos (2 x) - 3 cos (x) + 1 = 0

2cos^2(x)-cos(x)-1=0,0<= x<= 360

2 cos^{2} (x) - cos (x) - 1 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}