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4cos^3(x)=3cos(x)

解答

4 cos^{3} (x) = 3 cos (x)

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

4 cos^{3} (x) = 3 cos (x)

用替代法求解

4 cos^{3} (x) = 3 cos (x)

令

: cos (x) = u

4 u^{3} = 3 u

4 u^{3} = 3 u : u = 0, u = - \frac{3}{2}, u = \frac{3}{2}

4 u^{3} = 3 u

将

3 u

para o lado esquerdo

4 u^{3} = 3 u

两边减去

3 u

4 u^{3} - 3 u = 3 u - 3 u

化简

4 u^{3} - 3 u = 0

4 u^{3} - 3 u = 0

因式分解

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

4 u^{3} - 3 u

因式分解出通项

u : u (4 u^{2} - 3)

4 u^{3} - 3 u

使用指数法则:

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

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

= 4 u^{2} u - 3 u

因式分解出通项

u

= u (4 u^{2} - 3)

= u (4 u^{2} - 3)

分解

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

4 u^{2} - 3

将

4 u^{2} - 3

改写为

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

4 u^{2} - 3

将

4

改写为

2^{2}

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

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

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

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

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

= (2 u + 3) (2 u - 3)

= u (2 u + 3) (2 u - 3)

u (2 u + 3) (2 u - 3) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 2 u + 3 = 0 or 2 u - 3 = 0

解

2 u + 3 = 0 : u = - \frac{3}{2}

2 u + 3 = 0

将

3

到右边

2 u + 3 = 0

两边减去

3

2 u + 3 - 3 = 0 - 3

化简

2 u = - 3

2 u = - 3

两边除以

2

2 u = - 3

两边除以

2

\frac{2 u}{2} = \frac{- 3}{2}

化简

u = - \frac{3}{2}

u = - \frac{3}{2}

解

2 u - 3 = 0 : u = \frac{3}{2}

2 u - 3 = 0

将

3

到右边

2 u - 3 = 0

两边加上

3

2 u - 3 + 3 = 0 + 3

化简

2 u = 3

2 u = 3

两边除以

2

2 u = 3

两边除以

2

\frac{2 u}{2} = \frac{3}{2}

化简

u = \frac{3}{2}

u = \frac{3}{2}

解为

u = 0, u = - \frac{3}{2}, u = \frac{3}{2}

u = cos (x)

代回

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

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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