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受欢迎的三角函数 >

csc^5(θ)-4csc(θ)=0

解答

csc^{5} (θ) - 4 csc (θ) = 0

解答

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

+1

度数

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

求解步骤

csc^{5} (θ) - 4 csc (θ) = 0

用替代法求解

csc^{5} (θ) - 4 csc (θ) = 0

令

: csc (θ) = u

u^{5} - 4 u = 0

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

u^{5} - 4 u = 0

因式分解

u^{5} - 4 u : u (u^{2} + 2) (u + 2) (u - 2)

u^{5} - 4 u

因式分解出通项

u : u (u^{4} - 4)

u^{5} - 4 u

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{5} = u^{4} u

= u^{4} u - 4 u

因式分解出通项

u

= u (u^{4} - 4)

= u (u^{4} - 4)

分解

u^{4} - 4 : (u^{2} + 2) (u + 2) (u - 2)

u^{4} - 4

将

u^{4} - 4

改写为

(u^{2})^{2} - 2^{2}

u^{4} - 4

将

4

改写为

2^{2}

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

使用指数法则:

a^{b c} = (a^{b})^{c}

u^{4} = (u^{2})^{2}

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

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

使用平方差公式:

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

(u^{2})^{2} - 2^{2} = (u^{2} + 2) (u^{2} - 2)

= (u^{2} + 2) (u^{2} - 2)

分解

u^{2} - 2 : (u + 2) (u - 2)

u^{2} - 2

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用平方差公式:

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

u^{2} - (2)^{2} = (u + 2) (u - 2)

= (u + 2) (u - 2)

= (u^{2} + 2) (u + 2) (u - 2)

= u (u^{2} + 2) (u + 2) (u - 2)

u (u^{2} + 2) (u + 2) (u - 2) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or u^{2} + 2 = 0 or u + 2 = 0 or u - 2 = 0

解

u^{2} + 2 = 0 : u = 2 i, u = - 2 i

u^{2} + 2 = 0

将

2

到右边

u^{2} + 2 = 0

两边减去

2

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

化简

u^{2} = - 2

u^{2} = - 2

对于

x^{2} = f (a)

解为

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

u = - 2, u = - - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

化简

- - 2 : - 2 i

- - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

解

u + 2 = 0 : u = - 2

u + 2 = 0

将

2

到右边

u + 2 = 0

两边减去

2

u + 2 - 2 = 0 - 2

化简

u = - 2

u = - 2

解

u - 2 = 0 : u = 2

u - 2 = 0

将

2

到右边

u - 2 = 0

两边加上

2

u - 2 + 2 = 0 + 2

化简

u = 2

u = 2

解为

u = 0, u = 2 i, u = - 2 i, u = - 2, u = 2

u = csc (θ)

代回

csc (θ) = 0, csc (θ) = 2 i, csc (θ) = - 2 i, csc (θ) = - 2, csc (θ) = 2

csc (θ) = 0, csc (θ) = 2 i, csc (θ) = - 2 i, csc (θ) = - 2, csc (θ) = 2

csc (θ) = 0 :

无解

csc (θ) = 0

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

无解

csc (θ) = 2 i :

无解

csc (θ) = 2 i

无解

csc (θ) = - 2 i :

无解

csc (θ) = - 2 i

无解

csc (θ) = - 2 : θ = \frac{5 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

csc (θ) = - 2

csc (θ) = - 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

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

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

csc (θ) = 2 : θ = \frac{π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn

csc (θ) = 2

csc (θ) = 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

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

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

合并所有解

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

作图

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