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受欢迎的三角函数 >

(csc(x))/(cot(x))=csc(x)sec(x)

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

csc (x) sec (x)

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

化简

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

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

将项转换为分式:

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

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

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

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

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

分解

csc (x) - csc (x) sec (x) cot (x) : csc (x) (1 - sec (x) cot (x))

csc (x) - csc (x) sec (x) cot (x)

因式分解出通项

csc (x)

= csc (x) (1 - sec (x) cot (x))

csc (x) (1 - sec (x) cot (x)) = 0

分别求解每个部分

csc (x) = 0 or 1 - sec (x) cot (x) = 0

csc (x) = 0 :

无解

csc (x) = 0

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

无解

1 - sec (x) cot (x) = 0 :

无解

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

用 sin, cos 表示

1 - cot (x) sec (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

1 - \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 )}

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

将项转换为分式:

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

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

乘以:

1 \cdot sin (x) = sin (x)

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

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

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

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

- 1 + sin (x) = 0

将

1

到右边

- 1 + sin (x) = 0

两边加上

1

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

化简

sin (x) = 1

sin (x) = 1

sin (x) = 1

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

因为方程对以下值无定义:

\frac{π}{2} + 2 πn

无解

合并所有解

x \in R 无解

作图

流行的例子

sin(θ)= 1/(sqrt(3))

sin (θ) = \frac{1}{3}

cos (θ) = 0.9

4 - 4 sin (θ) = 4

4cos(x)sin(x)=-5sin(x)

4 cos (x) sin (x) = - 5 sin (x)

2sin(x)+cot(x)-csc(x)=0

2 sin (x) + cot (x) - csc (x) = 0