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受欢迎的三角函数 >

7cos(2x)=7cos(x)

解答

7 cos (2 x) = 7 cos (x)

解答

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

+1

度数

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

求解步骤

7 cos (2 x) = 7 cos (x)

两边减去

7 cos (x)

7 cos (2 x) - 7 cos (x) = 0

使用三角恒等式改写

7 cos (2 x) - 7 cos (x)

使用倍角公式:

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

= 7 (2 cos^{2} (x) - 1) - 7 cos (x)

(- 1 + 2 cos^{2} (x)) \cdot 7 - 7 cos (x) = 0

用替代法求解

(- 1 + 2 cos^{2} (x)) \cdot 7 - 7 cos (x) = 0

令

: cos (x) = u

(- 1 + 2 u^{2}) \cdot 7 - 7 u = 0

(- 1 + 2 u^{2}) \cdot 7 - 7 u = 0 : u = 1, u = - \frac{1}{2}

(- 1 + 2 u^{2}) \cdot 7 - 7 u = 0

展开

(- 1 + 2 u^{2}) \cdot 7 - 7 u : - 7 + 14 u^{2} - 7 u

(- 1 + 2 u^{2}) \cdot 7 - 7 u

= 7 (- 1 + 2 u^{2}) - 7 u

乘开

7 (- 1 + 2 u^{2}) : - 7 + 14 u^{2}

7 (- 1 + 2 u^{2})

使用分配律:

a (b + c) = ab + a c

a = 7, b = - 1, c = 2 u^{2}

= 7 (- 1) + 7 \cdot 2 u^{2}

使用加减运算法则

+ (- a) = - a

= - 7 \cdot 1 + 7 \cdot 2 u^{2}

化简

- 7 \cdot 1 + 7 \cdot 2 u^{2} : - 7 + 14 u^{2}

- 7 \cdot 1 + 7 \cdot 2 u^{2}

数字相乘:

7 \cdot 1 = 7

= - 7 + 7 \cdot 2 u^{2}

数字相乘:

7 \cdot 2 = 14

= - 7 + 14 u^{2}

= - 7 + 14 u^{2}

= - 7 + 14 u^{2} - 7 u

- 7 + 14 u^{2} - 7 u = 0

改写成标准形式

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

14 u^{2} - 7 u - 7 = 0

使用求根公式求解

14 u^{2} - 7 u - 7 = 0

二次方程求根公式:

若

a = 14, b = - 7, c = - 7

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 14 ( - 7 )}{2 \cdot 14}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 14 ( - 7 )}{2 \cdot 14}

(- 7)^{2} - 4 \cdot 14 (- 7) = 21

(- 7)^{2} - 4 \cdot 14 (- 7)

使用法则

- (- a) = a

= (- 7)^{2} + 4 \cdot 14 \cdot 7

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} + 4 \cdot 14 \cdot 7

数字相乘:

4 \cdot 14 \cdot 7 = 392

= 7^{2} + 392

7^{2} = 49

= 49 + 392

数字相加:

49 + 392 = 441

= 441

因式分解数字:

441 = 2 1^{2}

= 2 1^{2}

使用根式运算法则:

n a^{n} = a

2 1^{2} = 21

= 21

u_{1, 2} = \frac{- ( - 7 ) \pm 21}{2 \cdot 14}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 21}{2 \cdot 14}, u_{2} = \frac{- ( - 7 ) - 21}{2 \cdot 14}

u = \frac{- ( - 7 ) + 21}{2 \cdot 14} : 1

\frac{- ( - 7 ) + 21}{2 \cdot 14}

使用法则

- (- a) = a

= \frac{7 + 21}{2 \cdot 14}

数字相加:

7 + 21 = 28

= \frac{28}{2 \cdot 14}

数字相乘:

2 \cdot 14 = 28

= \frac{28}{28}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- ( - 7 ) - 21}{2 \cdot 14} : - \frac{1}{2}

\frac{- ( - 7 ) - 21}{2 \cdot 14}

使用法则

- (- a) = a

= \frac{7 - 21}{2 \cdot 14}

数字相减:

7 - 21 = - 14

= \frac{- 14}{2 \cdot 14}

数字相乘:

2 \cdot 14 = 28

= \frac{- 14}{28}

使用分式法则:

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

= - \frac{14}{28}

约分:

14

= - \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

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

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

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) = - \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

合并所有解

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

作图

流行的例子

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sin(1/2 x+pi/6)=(sqrt(3))/2

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