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受欢迎的三角函数 >

2sin(x)=cos(2x)

解答

2 sin (x) = cos (2 x)

解答

x = 0.37473 \dots + 2 πn, x = π - 0.37473 \dots + 2 πn

+1

度数

x = 21.47070 \dots^{\circ} + 36 0^{\circ} n, x = 158.52929 \dots^{\circ} + 36 0^{\circ} n

求解步骤

2 sin (x) = cos (2 x)

两边减去

cos (2 x)

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

使用三角恒等式改写

- cos (2 x) + 2 sin (x)

使用倍角公式:

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

= - (1 - 2 sin^{2} (x)) + 2 sin (x)

- (1 - 2 sin^{2} (x)) : - 1 + 2 sin^{2} (x)

- (1 - 2 sin^{2} (x))

打开括号

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

使用加减运算法则

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

= - 1 + 2 sin^{2} (x)

= - 1 + 2 sin^{2} (x) + 2 sin (x)

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

用替代法求解

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

令

: sin (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

2^{2} - 4 \cdot 2 (- 1) = 23

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

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 2^{2} + 8

2^{2} = 4

= 4 + 8

数字相加:

4 + 8 = 12

= 12

12

质因数分解:

2^{2} \cdot 3

12

12

除以

212 = 6 \cdot 2

= 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

使用根式运算法则:

n ab = n a n b

= 3 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 23

u_{1, 2} = \frac{- 2 \pm 2 3}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 2 + 2 3}{2 \cdot 2}, u_{2} = \frac{- 2 - 2 3}{2 \cdot 2}

u = \frac{- 2 + 2 3}{2 \cdot 2} : \frac{- 1 + 3}{2}

\frac{- 2 + 2 3}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 2 + 2 3}{4}

分解

- 2 + 23 : 2 (- 1 + 3)

- 2 + 23

改写为

= - 2 \cdot 1 + 23

因式分解出通项

2

= 2 (- 1 + 3)

= \frac{2 ( - 1 + 3 )}{4}

约分:

2

= \frac{- 1 + 3}{2}

u = \frac{- 2 - 2 3}{2 \cdot 2} : - \frac{1 + 3}{2}

\frac{- 2 - 2 3}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 2 - 2 3}{4}

分解

- 2 - 23 : - 2 (1 + 3)

- 2 - 23

改写为

= - 2 \cdot 1 - 23

因式分解出通项

2

= - 2 (1 + 3)

= - \frac{2 ( 1 + 3 )}{4}

约分:

2

= - \frac{1 + 3}{2}

二次方程组的解是:

u = \frac{- 1 + 3}{2}, u = - \frac{1 + 3}{2}

u = sin (x)

代回

sin (x) = \frac{- 1 + 3}{2}, sin (x) = - \frac{1 + 3}{2}

sin (x) = \frac{- 1 + 3}{2}, sin (x) = - \frac{1 + 3}{2}

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

sin (x) = \frac{- 1 + 3}{2}

使用反三角函数性质

sin (x) = \frac{- 1 + 3}{2}

sin (x) = \frac{- 1 + 3}{2}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

sin (x) = - \frac{1 + 3}{2} :

无解

sin (x) = - \frac{1 + 3}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

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

以小数形式表示解

x = 0.37473 \dots + 2 πn, x = π - 0.37473 \dots + 2 πn

作图

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