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受欢迎的三角函数 >

3cos(x)=3cos(2x)

解答

3 cos (x) = 3 cos (2 x)

解答

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 2 πn

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n

求解步骤

3 cos (x) = 3 cos (2 x)

两边减去

3 cos (2 x)

3 cos (x) - 3 cos (2 x) = 0

使用三角恒等式改写

- 3 cos (2 x) + 3 cos (x)

使用倍角公式:

cos (2 x) = 2 cos^{2} (x) - 1

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

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

用替代法求解

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

令

: cos (x) = u

- (- 1 + 2 u^{2}) \cdot 3 + 3 u = 0

- (- 1 + 2 u^{2}) \cdot 3 + 3 u = 0 : u = - \frac{1}{2}, u = 1

- (- 1 + 2 u^{2}) \cdot 3 + 3 u = 0

展开

- (- 1 + 2 u^{2}) \cdot 3 + 3 u : 3 - 6 u^{2} + 3 u

- (- 1 + 2 u^{2}) \cdot 3 + 3 u

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

乘开

- 3 (- 1 + 2 u^{2}) : 3 - 6 u^{2}

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

使用分配律:

a (b + c) = ab + a c

a = - 3, b = - 1, c = 2 u^{2}

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

使用加减运算法则

- (- a) = a, + (- a) = - a

= 3 \cdot 1 - 3 \cdot 2 u^{2}

化简

3 \cdot 1 - 3 \cdot 2 u^{2} : 3 - 6 u^{2}

3 \cdot 1 - 3 \cdot 2 u^{2}

数字相乘:

3 \cdot 1 = 3

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

数字相乘:

3 \cdot 2 = 6

= 3 - 6 u^{2}

= 3 - 6 u^{2}

= 3 - 6 u^{2} + 3 u

3 - 6 u^{2} + 3 u = 0

改写成标准形式

a x^{2} + b x + c = 0

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 6, b = 3, c = 3

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

3^{2} - 4 (- 6) \cdot 3 = 9

3^{2} - 4 (- 6) \cdot 3

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 6 \cdot 3

数字相乘:

4 \cdot 6 \cdot 3 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

数字相加:

9 + 72 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 3 \pm 9}{2 ( - 6 )}

将解分隔开

u_{1} = \frac{- 3 + 9}{2 ( - 6 )}, u_{2} = \frac{- 3 - 9}{2 ( - 6 )}

u = \frac{- 3 + 9}{2 ( - 6 )} : - \frac{1}{2}

\frac{- 3 + 9}{2 ( - 6 )}

去除括号:

(- a) = - a

= \frac{- 3 + 9}{- 2 \cdot 6}

数字相加/相减:

- 3 + 9 = 6

= \frac{6}{- 2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{6}{- 12}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6}{12}

约分:

6

= - \frac{1}{2}

u = \frac{- 3 - 9}{2 ( - 6 )} : 1

\frac{- 3 - 9}{2 ( - 6 )}

去除括号:

(- a) = - a

= \frac{- 3 - 9}{- 2 \cdot 6}

数字相减:

- 3 - 9 = - 12

= \frac{- 12}{- 2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{- 12}{- 12}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{12}{12}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

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

u = cos (x)

代回

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

合并所有解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 2 πn

作图

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