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受欢迎的三角函数 >

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

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

sec (x)

\frac{cos ( x )}{cot ( x )} - sec (x) = 0

化简

\frac{cos ( x )}{cot ( x )} - sec (x) : \frac{cos ( x ) - sec ( x ) cot ( x )}{cot ( x )}

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

将项转换为分式:

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

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

\frac{cos ( x ) - sec ( x ) cot ( x )}{cot ( x )} = 0

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

cos (x) - sec (x) cot (x) = 0

用 sin, cos 表示

cos (x) - cot (x) sec (x)

使用基本三角恒等式:

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

= cos (x) - \frac{cos ( x )}{sin ( x )} sec (x)

使用基本三角恒等式:

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

= cos (x) - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

化简

cos (x) - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{cos ( x ) sin ( x ) - 1}{sin ( x )}

cos (x) - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} = \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

分式相乘:

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

= \frac{cos ( x ) \cdot 1}{sin ( x ) cos ( x )}

约分:

cos (x)

= \frac{1}{sin ( x )}

= cos (x) - \frac{1}{sin ( x )}

将项转换为分式:

cos (x) = \frac{cos ( x ) sin ( x )}{sin ( x )}

= \frac{cos ( x ) sin ( x )}{sin ( x )} - \frac{1}{sin ( 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{cos ( x ) sin ( x ) - 1}{sin ( x )}

= \frac{cos ( x ) sin ( x ) - 1}{sin ( x )}

\frac{- 1 + cos ( x ) sin ( x )}{sin ( x )} = 0

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

- 1 + cos (x) sin (x) = 0

使用三角恒等式改写

- 1 + cos (x) sin (x)

使用倍角公式:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 1 + \frac{sin ( 2 x )}{2}

- 1 + \frac{sin ( 2 x )}{2} = 0

将

1

到右边

- 1 + \frac{sin ( 2 x )}{2} = 0

两边加上

1

- 1 + \frac{sin ( 2 x )}{2} + 1 = 0 + 1

化简

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

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

在两边乘以

2

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

在两边乘以

2

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

化简

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

x \in R 无解

作图

流行的例子

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cos(2x)-sin(-x)=0

cos (2 x) - sin (- x) = 0

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(4cos^2(x)-3)(csc(x)+2)=0

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