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受欢迎的三角函数 >

sec^5(x)=4sec(x)

解答

sec^{5} (x) = 4 sec (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

sec^{5} (x) = 4 sec (x)

用替代法求解

sec^{5} (x) = 4 sec (x)

令

: sec (x) = u

u^{5} = 4 u

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

u^{5} = 4 u

将

4 u

para o lado esquerdo

u^{5} = 4 u

两边减去

4 u

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

化简

u^{5} - 4 u = 0

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 = sec (x)

代回

sec (x) = 0, sec (x) = 2 i, sec (x) = - 2 i, sec (x) = - 2, sec (x) = 2

sec (x) = 0, sec (x) = 2 i, sec (x) = - 2 i, sec (x) = - 2, sec (x) = 2

sec (x) = 0 :

无解

sec (x) = 0

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

无解

sec (x) = 2 i :

无解

sec (x) = 2 i

无解

sec (x) = - 2 i :

无解

sec (x) = - 2 i

无解

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) = - 2

sec (x) = - 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

sec (x) = 2

sec (x) = 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

- 2 = 4 sin (3 x)

7cos(x)=3+cos(x)

7 cos (x) = 3 + cos (x)

sin (t) = \frac{3}{4}

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cos (2 θ) = \frac{- 3}{2}

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