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受欢迎的三角函数 >

cos^4(x)-1=0

解答

cos^{4} (x) - 1 = 0

解答

x = 2 πn, x = π + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

cos^{4} (x) - 1 = 0

用替代法求解

cos^{4} (x) - 1 = 0

令

: cos (x) = u

u^{4} - 1 = 0

u^{4} - 1 = 0 : u = 1, u = - 1, u = i, u = - i

u^{4} - 1 = 0

将

1

到右边

u^{4} - 1 = 0

两边加上

1

u^{4} - 1 + 1 = 0 + 1

化简

u^{4} = 1

u^{4} = 1

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} = 1

解

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

v^{2} = 1

对于

(g (x))^{2} = f (a)

解为

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

v = 1, v = - 1

v = 1, v = - 1

代回

v = u^{2}

，求解

u

解

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

u^{2} = 1

使用法则

1 = 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

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

u^{2} = - 1

使用法则

1 = 1

u^{2} = - 1

对于

x^{2} = f (a)

解为

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

u = - 1, u = - - 1

化简

- 1 : i

- 1

使用虚数运算法则:

- 1 = i

= i

化简

- - 1 : - i

- - 1

使用虚数运算法则:

- 1 = i

= - i

u = i, u = - i

解为

u = 1, u = - 1, u = i, u = - i

u = cos (x)

代回

cos (x) = 1, cos (x) = - 1, cos (x) = i, cos (x) = - i

cos (x) = 1, cos (x) = - 1, cos (x) = i, cos (x) = - i

cos (x) = 1 : x = 2 πn

cos (x) = 1

cos (x) = 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

cos (x) = i :

无解

cos (x) = i

无解

cos (x) = - i :

无解

cos (x) = - i

无解

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

2sin(x)=sqrt(3),0<= x<2pi

sec^2(x)+0.5tan(x)=1

8sin(x)=cos(x)-3

cos(2x)-sin^2(x)=1