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受欢迎的三角函数 >

cot^2(x)-2csc^2(x)+5=0

解答

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

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

+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

求解步骤

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

使用三角恒等式改写

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

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

化简

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

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

对同类项分组

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

同类项相加:

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

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

数字相加/相减:

5 - 1 = 4

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

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

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

用替代法求解

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

令

: csc (x) = u

4 - u^{2} = 0

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

4 - u^{2} = 0

将

4

到右边

4 - u^{2} = 0

两边减去

4

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

化简

- u^{2} = - 4

- u^{2} = - 4

两边除以

- 1

- u^{2} = - 4

两边除以

- 1

\frac{- u ^{2}}{- 1} = \frac{- 4}{- 1}

化简

u^{2} = 4

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

u = csc (x)

代回

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

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

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

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

作图

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