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受欢迎的三角函数 >

4sin(x)-cos(2x)=1

解答

4 sin (x) - cos (2 x) = 1

解答

x = 0.42707 \dots + 2 πn, x = π - 0.42707 \dots + 2 πn

+1

度数

x = 24.46980 \dots^{\circ} + 36 0^{\circ} n, x = 155.53019 \dots^{\circ} + 36 0^{\circ} n

求解步骤

4 sin (x) - cos (2 x) = 1

两边减去

1

4 sin (x) - cos (2 x) - 1 = 0

使用三角恒等式改写

- 1 - cos (2 x) + 4 sin (x)

使用倍角公式:

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

= - 1 - (1 - 2 sin^{2} (x)) + 4 sin (x)

化简

- 1 - (1 - 2 sin^{2} (x)) + 4 sin (x) : 2 sin^{2} (x) + 4 sin (x) - 2

- 1 - (1 - 2 sin^{2} (x)) + 4 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 - 1 + 2 sin^{2} (x) + 4 sin (x)

数字相减:

- 1 - 1 = - 2

= 2 sin^{2} (x) + 4 sin (x) - 2

= 2 sin^{2} (x) + 4 sin (x) - 2

- 2 + 2 sin^{2} (x) + 4 sin (x) = 0

用替代法求解

- 2 + 2 sin^{2} (x) + 4 sin (x) = 0

令

: sin (x) = u

- 2 + 2 u^{2} + 4 u = 0

- 2 + 2 u^{2} + 4 u = 0 : u = - 1 + 2, u = - 1 - 2

- 2 + 2 u^{2} + 4 u = 0

改写成标准形式

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

2 u^{2} + 4 u - 2 = 0

使用求根公式求解

2 u^{2} + 4 u - 2 = 0

二次方程求根公式:

若

a = 2, b = 4, c = - 2

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

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

4^{2} - 4 \cdot 2 (- 2) = 42

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 4^{2} + 16

4^{2} = 16

= 16 + 16

数字相加:

16 + 16 = 32

= 32

32

质因数分解:

2^{5}

32

32

除以

232 = 16 \cdot 2

= 2 \cdot 16

16

除以

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{5}

= 2^{5}

使用指数法则:

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

= 2^{4} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{4}

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}}

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2

整理后得

= 42

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

将解分隔开

u_{1} = \frac{- 4 + 4 2}{2 \cdot 2}, u_{2} = \frac{- 4 - 4 2}{2 \cdot 2}

u = \frac{- 4 + 4 2}{2 \cdot 2} : - 1 + 2

\frac{- 4 + 4 2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

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

分解

- 4 + 42 : 4 (- 1 + 2)

- 4 + 42

改写为

= - 4 \cdot 1 + 42

因式分解出通项

4

= 4 (- 1 + 2)

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

数字相除:

\frac{4}{4} = 1

= - 1 + 2

u = \frac{- 4 - 4 2}{2 \cdot 2} : - 1 - 2

\frac{- 4 - 4 2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

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

分解

- 4 - 42 : - 4 (1 + 2)

- 4 - 42

改写为

= - 4 \cdot 1 - 42

因式分解出通项

4

= - 4 (1 + 2)

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

数字相除:

\frac{4}{4} = 1

= - (1 + 2)

取负

- (1 + 2) = - 1 - 2

= - 1 - 2

二次方程组的解是:

u = - 1 + 2, u = - 1 - 2

u = sin (x)

代回

sin (x) = - 1 + 2, sin (x) = - 1 - 2

sin (x) = - 1 + 2, sin (x) = - 1 - 2

sin (x) = - 1 + 2 : x = arcsin (- 1 + 2) + 2 πn, x = π - arcsin (- 1 + 2) + 2 πn

sin (x) = - 1 + 2

使用反三角函数性质

sin (x) = - 1 + 2

sin (x) = - 1 + 2

的通解

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

x = arcsin (- 1 + 2) + 2 πn, x = π - arcsin (- 1 + 2) + 2 πn

x = arcsin (- 1 + 2) + 2 πn, x = π - arcsin (- 1 + 2) + 2 πn

sin (x) = - 1 - 2 :

无解

sin (x) = - 1 - 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (- 1 + 2) + 2 πn, x = π - arcsin (- 1 + 2) + 2 πn

以小数形式表示解

x = 0.42707 \dots + 2 πn, x = π - 0.42707 \dots + 2 πn

作图

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