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受欢迎的三角函数 >

cos(2x)=cos(x)-cos(30)[180]

解答

cos (2 x) = cos (x) - cos (3 0^{\circ}) [180]

解答

x \in R 无解

求解步骤

cos (2 x) = cos (x) - cos (3 0^{\circ}) [180]

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

使用以下普通恒等式:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2}

cos (2 x) = cos (x) - \frac{3}{2} [180]

两边减去

cos (x) - \frac{3}{2} [180]

cos (2 x) - cos (x) + 903 = 0

使用三角恒等式改写

cos (2 x) - cos (x) + 903

使用倍角公式:

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

= 2 cos^{2} (x) - 1 - cos (x) + 903

- 1 - cos (x) + 2 cos^{2} (x) + 903 = 0

用替代法求解

- 1 - cos (x) + 2 cos^{2} (x) + 903 = 0

令

: cos (x) = u

- 1 - u + 2 u^{2} + 903 = 0

- 1 - u + 2 u^{2} + 903 = 0 : u = \frac{1}{4} + i \frac{3 - 1 + 80 3}{4}, u = \frac{1}{4} - i \frac{3 - 1 + 80 3}{4}

- 1 - u + 2 u^{2} + 903 = 0

改写成标准形式

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

2 u^{2} - u - 1 + 903 = 0

使用求根公式求解

2 u^{2} - u - 1 + 903 = 0

二次方程求根公式:

若

a = 2, b = - 1, c = - 1 + 903

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 ( - 1 + 90 3 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 ( - 1 + 90 3 )}{2 \cdot 2}

化简

(- 1)^{2} - 4 \cdot 2 (- 1 + 903) : 3 i 803 - 1

(- 1)^{2} - 4 \cdot 2 (- 1 + 903)

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 2 (- 1 + 903) = 8 (- 1 + 903)

4 \cdot 2 (- 1 + 903)

数字相乘:

4 \cdot 2 = 8

= 8 (903 - 1)

= 1 - 8 (903 - 1)

使用虚数运算法则:

- a = i a

= i 8 (903 - 1) - 1

- 1 + 8 (- 1 + 903) = 3 803 - 1

- 1 + 8 (- 1 + 903)

乘开

- 1 + 8 (- 1 + 903) : 7203 - 9

- 1 + 8 (- 1 + 903)

乘开

8 (- 1 + 903) : - 8 + 7203

8 (- 1 + 903)

使用分配律:

a (b + c) = ab + a c

a = 8, b = - 1, c = 903

= 8 (- 1) + 8 \cdot 903

使用加减运算法则

+ (- a) = - a

= - 8 \cdot 1 + 8 \cdot 903

化简

- 8 \cdot 1 + 8 \cdot 903 : - 8 + 7203

- 8 \cdot 1 + 8 \cdot 903

数字相乘:

8 \cdot 1 = 8

= - 8 + 8 \cdot 903

数字相乘:

8 \cdot 90 = 720

= - 8 + 7203

= - 8 + 7203

= - 1 - 8 + 7203

数字相减:

- 1 - 8 = - 9

= 7203 - 9

= 7203 - 9

分解

7203 - 9 : 9 (803 - 1)

7203 - 9

改写为

= 9 \cdot 803 - 9 \cdot 1

因式分解出通项

9

= 9 (803 - 1)

= 9 (803 - 1)

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= 9 803 - 1

9 = 3

9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

= 3 803 - 1

= 3 i 803 - 1

u_{1, 2} = \frac{- ( - 1 ) \pm 3 i 80 3 - 1}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 3 i 80 3 - 1}{2 \cdot 2}, u_{2} = \frac{- ( - 1 ) - 3 i 80 3 - 1}{2 \cdot 2}

u = \frac{- ( - 1 ) + 3 i 80 3 - 1}{2 \cdot 2} : \frac{1}{4} + i \frac{3 - 1 + 80 3}{4}

\frac{- ( - 1 ) + 3 i 80 3 - 1}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{1 + 3 i 80 3 - 1}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{1 + 3 i 80 3 - 1}{4}

将

\frac{1 + 3 i 80 3 - 1}{4}

改写成标准复数形式:

\frac{1}{4} + \frac{3 80 3 - 1}{4} i

\frac{1 + 3 i 80 3 - 1}{4}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 + 3 i 80 3 - 1}{4} = \frac{1}{4} + \frac{3 i 80 3 - 1}{4}

= \frac{1}{4} + \frac{3 i 80 3 - 1}{4}

= \frac{1}{4} + \frac{3 80 3 - 1}{4} i

u = \frac{- ( - 1 ) - 3 i 80 3 - 1}{2 \cdot 2} : \frac{1}{4} - i \frac{3 - 1 + 80 3}{4}

\frac{- ( - 1 ) - 3 i 80 3 - 1}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{1 - 3 i 80 3 - 1}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{1 - 3 i 80 3 - 1}{4}

将

\frac{1 - 3 i 80 3 - 1}{4}

改写成标准复数形式:

\frac{1}{4} - \frac{3 80 3 - 1}{4} i

\frac{1 - 3 i 80 3 - 1}{4}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 - 3 i 80 3 - 1}{4} = \frac{1}{4} - \frac{3 i 80 3 - 1}{4}

= \frac{1}{4} - \frac{3 i 80 3 - 1}{4}

= \frac{1}{4} - \frac{3 80 3 - 1}{4} i

二次方程组的解是:

u = \frac{1}{4} + i \frac{3 - 1 + 80 3}{4}, u = \frac{1}{4} - i \frac{3 - 1 + 80 3}{4}

u = cos (x)

代回

cos (x) = \frac{1}{4} + i \frac{3 - 1 + 80 3}{4}, cos (x) = \frac{1}{4} - i \frac{3 - 1 + 80 3}{4}

cos (x) = \frac{1}{4} + i \frac{3 - 1 + 80 3}{4}, cos (x) = \frac{1}{4} - i \frac{3 - 1 + 80 3}{4}

cos (x) = \frac{1}{4} + i \frac{3 - 1 + 80 3}{4} :

无解

cos (x) = \frac{1}{4} + i \frac{3 - 1 + 80 3}{4}

无解

cos (x) = \frac{1}{4} - i \frac{3 - 1 + 80 3}{4} :

无解

cos (x) = \frac{1}{4} - i \frac{3 - 1 + 80 3}{4}

无解

合并所有解

x \in R 无解

作图

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