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受欢迎的三角函数 >

2cos^3(x)-cos^2(x)+2cos(x)-1=0

解答

2 cos^{3} (x) - cos^{2} (x) + 2 cos (x) - 1 = 0

解答

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

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

2 cos^{3} (x) - cos^{2} (x) + 2 cos (x) - 1 = 0

用替代法求解

2 cos^{3} (x) - cos^{2} (x) + 2 cos (x) - 1 = 0

令

: cos (x) = u

2 u^{3} - u^{2} + 2 u - 1 = 0

2 u^{3} - u^{2} + 2 u - 1 = 0 : u = \frac{1}{2}, u = i, u = - i

2 u^{3} - u^{2} + 2 u - 1 = 0

因式分解

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

2 u^{3} - u^{2} + 2 u - 1

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

从

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

分解出因式

u^{2}

:

u^{2} (2 u - 1)

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

使用指数法则:

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

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

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

因式分解出通项

u^{2}

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

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

因式分解出通项

2 u - 1

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

(2 u - 1) (u^{2} + 1) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

2 u - 1 = 0 or u^{2} + 1 = 0

解

2 u - 1 = 0 : u = \frac{1}{2}

2 u - 1 = 0

将

1

到右边

2 u - 1 = 0

两边加上

1

2 u - 1 + 1 = 0 + 1

化简

2 u = 1

2 u = 1

两边除以

2

2 u = 1

两边除以

2

\frac{2 u}{2} = \frac{1}{2}

化简

u = \frac{1}{2}

u = \frac{1}{2}

解

u^{2} + 1 = 0 : u = i, u = - i

u^{2} + 1 = 0

将

1

到右边

u^{2} + 1 = 0

两边减去

1

u^{2} + 1 - 1 = 0 - 1

化简

u^{2} = - 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 = \frac{1}{2}, u = i, u = - i

u = cos (x)

代回

cos (x) = \frac{1}{2}, cos (x) = i, cos (x) = - i

cos (x) = \frac{1}{2}, cos (x) = i, cos (x) = - i

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

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

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

作图

流行的例子

sin (x) = \frac{2 π}{3}

cos(3x)+sin(2x)+cos(x)=0

cos (3 x) + sin (2 x) + cos (x) = 0

tan (2 θ) = 1.333

2 sec (x) + 4 = 0

2cos(x)=3sin(x)

2 cos (x) = 3 sin (x)