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受欢迎的三角函数 >

csc^3(x)+csc^2(x)=2csc(x)+2

解答

csc^{3} (x) + csc^{2} (x) = 2 csc (x) + 2

解答

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

+1

度数

x = 27 0^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

求解步骤

csc^{3} (x) + csc^{2} (x) = 2 csc (x) + 2

用替代法求解

csc^{3} (x) + csc^{2} (x) = 2 csc (x) + 2

令

: csc (x) = u

u^{3} + u^{2} = 2 u + 2

u^{3} + u^{2} = 2 u + 2 : u = - 1, u = - 2, u = 2

u^{3} + u^{2} = 2 u + 2

将

2

para o lado esquerdo

u^{3} + u^{2} = 2 u + 2

两边减去

2

u^{3} + u^{2} - 2 = 2 u + 2 - 2

化简

u^{3} + u^{2} - 2 = 2 u

u^{3} + u^{2} - 2 = 2 u

将

2 u

para o lado esquerdo

u^{3} + u^{2} - 2 = 2 u

两边减去

2 u

u^{3} + u^{2} - 2 - 2 u = 2 u - 2 u

化简

u^{3} + u^{2} - 2 - 2 u = 0

u^{3} + u^{2} - 2 - 2 u = 0

改写成标准形式

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

u^{3} + u^{2} - 2 u - 2 = 0

因式分解

u^{3} + u^{2} - 2 u - 2 : (u + 1) (u + 2) (u - 2)

u^{3} + u^{2} - 2 u - 2

= (u^{3} + u^{2}) + (- 2 u - 2)

从

- 2 u - 2

分解出因式

- 2

:

- 2 (u + 1)

- 2 u - 2

因式分解出通项

- 2

= - 2 (u + 1)

从

u^{3} + u^{2}

分解出因式

u^{2}

:

u^{2} (u + 1)

u^{3} + u^{2}

使用指数法则:

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

u^{3} = u u^{2}

= u u^{2} + u^{2}

因式分解出通项

u^{2}

= u^{2} (u + 1)

= - 2 (u + 1) + u^{2} (u + 1)

因式分解出通项

u + 1

= (u + 1) (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 + 1) (u + 2) (u - 2)

(u + 1) (u + 2) (u - 2) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u + 1 = 0 or u + 2 = 0 or u - 2 = 0

解

u + 1 = 0 : u = - 1

u + 1 = 0

将

1

到右边

u + 1 = 0

两边减去

1

u + 1 - 1 = 0 - 1

化简

u = - 1

u = - 1

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

u = csc (x)

代回

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

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

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

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

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

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

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

合并所有解

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

作图

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