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受欢迎的三角函数 >

csc^4(x)-1=5cot^2(x)

解答

csc^{4} (x) - 1 = 5 cot^{2} (x)

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

csc^{4} (x) - 1 = 5 cot^{2} (x)

两边减去

5 cot^{2} (x)

csc^{4} (x) - 1 - 5 cot^{2} (x) = 0

使用三角恒等式改写

- 1 + csc^{4} (x) - 5 cot^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 1 + csc^{4} (x) - 5 (csc^{2} (x) - 1)

化简

- 1 + csc^{4} (x) - 5 (csc^{2} (x) - 1) : csc^{4} (x) - 5 csc^{2} (x) + 4

- 1 + csc^{4} (x) - 5 (csc^{2} (x) - 1)

乘开

- 5 (csc^{2} (x) - 1) : - 5 csc^{2} (x) + 5

- 5 (csc^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = - 5, b = csc^{2} (x), c = 1

= - 5 csc^{2} (x) - (- 5) \cdot 1

使用加减运算法则

- (- a) = a

= - 5 csc^{2} (x) + 5 \cdot 1

数字相乘:

5 \cdot 1 = 5

= - 5 csc^{2} (x) + 5

= - 1 + csc^{4} (x) - 5 csc^{2} (x) + 5

化简

- 1 + csc^{4} (x) - 5 csc^{2} (x) + 5 : csc^{4} (x) - 5 csc^{2} (x) + 4

- 1 + csc^{4} (x) - 5 csc^{2} (x) + 5

对同类项分组

= csc^{4} (x) - 5 csc^{2} (x) - 1 + 5

数字相加/相减:

- 1 + 5 = 4

= csc^{4} (x) - 5 csc^{2} (x) + 4

= csc^{4} (x) - 5 csc^{2} (x) + 4

= csc^{4} (x) - 5 csc^{2} (x) + 4

4 + csc^{4} (x) - 5 csc^{2} (x) = 0

用替代法求解

4 + csc^{4} (x) - 5 csc^{2} (x) = 0

令

: csc (x) = u

4 + u^{4} - 5 u^{2} = 0

4 + u^{4} - 5 u^{2} = 0 : u = 2, u = - 2, u = 1, u = - 1

4 + u^{4} - 5 u^{2} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - 5 u^{2} + 4 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} - 5 v + 4 = 0

解

v^{2} - 5 v + 4 = 0 : v = 4, v = 1

v^{2} - 5 v + 4 = 0

使用求根公式求解

v^{2} - 5 v + 4 = 0

二次方程求根公式:

若

a = 1, b = - 5, c = 4

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 4}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot 4 = 3

(- 5)^{2} - 4 \cdot 1 \cdot 4

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 1 \cdot 4

数字相乘:

4 \cdot 1 \cdot 4 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

数字相减:

25 - 16 = 9

= 9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

v_{1, 2} = \frac{- ( - 5 ) \pm 3}{2 \cdot 1}

将解分隔开

v_{1} = \frac{- ( - 5 ) + 3}{2 \cdot 1}, v_{2} = \frac{- ( - 5 ) - 3}{2 \cdot 1}

v = \frac{- ( - 5 ) + 3}{2 \cdot 1} : 4

\frac{- ( - 5 ) + 3}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{5 + 3}{2 \cdot 1}

数字相加:

5 + 3 = 8

= \frac{8}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8}{2}

数字相除:

\frac{8}{2} = 4

= 4

v = \frac{- ( - 5 ) - 3}{2 \cdot 1} : 1

\frac{- ( - 5 ) - 3}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{5 - 3}{2 \cdot 1}

数字相减:

5 - 3 = 2

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

数字相乘:

2 \cdot 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

v = 4, v = 1

v = 4, v = 1

代回

v = u^{2}

，求解

u

解

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = 4, u = - 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

解

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

使用法则

1 = 1

= 1

- 1 = - 1

- 1

使用法则

1 = 1

= - 1

u = 1, u = - 1

解为

u = 2, u = - 2, u = 1, u = - 1

u = csc (x)

代回

csc (x) = 2, csc (x) = - 2, csc (x) = 1, csc (x) = - 1

csc (x) = 2, csc (x) = - 2, csc (x) = 1, csc (x) = - 1

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

csc (x) = 2

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = - 2

csc (x) = - 2

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

csc (x) = 1

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

csc (x) = - 1

csc (x) = - 1

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

流行的例子

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