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受欢迎的三角函数 >

csc^2(x)=2*cot^2(x)

解答

csc^{2} (x) = 2 \cdot cot^{2} (x)

解答

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

csc^{2} (x) = 2 cot^{2} (x)

两边减去

2 cot^{2} (x)

csc^{2} (x) - 2 cot^{2} (x) = 0

分解

csc^{2} (x) - 2 cot^{2} (x) : (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

csc^{2} (x) - 2 cot^{2} (x)

将

csc^{2} (x) - 2 cot^{2} (x)

改写为

csc^{2} (x) - (2 cot (x))^{2}

csc^{2} (x) - 2 cot^{2} (x)

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

= csc^{2} (x) - (2)^{2} cot^{2} (x)

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(2)^{2} cot^{2} (x) = (2 cot (x))^{2}

= csc^{2} (x) - (2 cot (x))^{2}

= csc^{2} (x) - (2 cot (x))^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

分别求解每个部分

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

csc (x) + 2 cot (x) = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

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

用 sin, cos 表示

csc (x) + cot (x) 2

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

\frac{1}{sin ( x )} + \frac{cos ( x )}{sin ( x )} 2 : \frac{1 + 2 cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} + \frac{cos ( x )}{sin ( x )} 2

乘

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

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

分式相乘:

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

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

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

使用法则

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

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

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

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

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

1 + cos (x) 2 = 0

将

1

到右边

1 + cos (x) 2 = 0

两边减去

1

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

化简

cos (x) 2 = - 1

cos (x) 2 = - 1

两边除以

2

cos (x) 2 = - 1

两边除以

2

\frac{cos ( x ) 2}{2} = \frac{- 1}{2}

化简

\frac{cos ( x ) 2}{2} = \frac{- 1}{2}

化简

\frac{cos ( x ) 2}{2} : cos (x)

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

约分:

2

= cos (x)

化简

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

使用分式法则:

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

= - \frac{1}{2}

- \frac{1}{2}

有理化:

- \frac{2}{2}

- \frac{1}{2}

乘以共轭根式

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

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

的通解

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

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

csc (x) - 2 cot (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

用 sin, cos 表示

csc (x) - cot (x) 2

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} 2 : \frac{1 - 2 cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} 2

乘

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

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

分式相乘:

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

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

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

使用法则

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

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

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

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

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

1 - cos (x) 2 = 0

将

1

到右边

1 - cos (x) 2 = 0

两边减去

1

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

化简

- cos (x) 2 = - 1

- cos (x) 2 = - 1

两边除以

- 2

- cos (x) 2 = - 1

两边除以

- 2

\frac{- cos ( x ) 2}{- 2} = \frac{- 1}{- 2}

化简

\frac{- cos ( x ) 2}{- 2} = \frac{- 1}{- 2}

化简

\frac{- cos ( x ) 2}{- 2} : cos (x)

\frac{- cos ( x ) 2}{- 2}

使用分式法则:

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

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

约分:

2

= cos (x)

化简

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

使用分式法则:

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

= \frac{1}{2}

\frac{1}{2}

有理化:

\frac{2}{2}

\frac{1}{2}

乘以共轭根式

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

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

的通解

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

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

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tan(x)=sqrt(3),-180<x<180

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cot^2(t)csc(t)=sin(t)

cot^{2} (t) csc (t) = sin (t)

\frac{6}{12} = cos (θ)

sin(a)=(6.88)/(6.9)

sin (a) = \frac{6.88}{6.9}