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受欢迎的三角函数 >

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

解答

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

解答

x = - 0.43939 \dots + 2 πn, x = π + 0.43939 \dots + 2 πn

+1

度数

x = - 25.17538 \dots^{\circ} + 36 0^{\circ} n, x = 205.17538 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 2 cos (2 x) - 3 sin (x)

使用倍角公式:

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

= - 2 (1 - 2 sin^{2} (x)) - 3 sin (x)

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

用替代法求解

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

令

: sin (x) = u

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

- (1 - 2 u^{2}) \cdot 2 - 3 u = 0 : u = \frac{3 + 41}{8}, u = \frac{3 - 41}{8}

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

展开

- (1 - 2 u^{2}) \cdot 2 - 3 u : - 2 + 4 u^{2} - 3 u

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

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

乘开

- 2 (1 - 2 u^{2}) : - 2 + 4 u^{2}

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

使用分配律:

a (b - c) = ab - a c

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

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

使用加减运算法则

- (- a) = a

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

化简

- 2 \cdot 1 + 2 \cdot 2 u^{2} : - 2 + 4 u^{2}

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

数字相乘:

2 \cdot 1 = 2

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

数字相乘:

2 \cdot 2 = 4

= - 2 + 4 u^{2}

= - 2 + 4 u^{2}

= - 2 + 4 u^{2} - 3 u

- 2 + 4 u^{2} - 3 u = 0

改写成标准形式

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

4 u^{2} - 3 u - 2 = 0

使用求根公式求解

4 u^{2} - 3 u - 2 = 0

二次方程求根公式:

若

a = 4, b = - 3, c = - 2

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

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

(- 3)^{2} - 4 \cdot 4 (- 2) = 41

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

(- 3)^{2} = 3^{2}

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

数字相乘:

4 \cdot 4 \cdot 2 = 32

= 3^{2} + 32

3^{2} = 9

= 9 + 32

数字相加:

9 + 32 = 41

= 41

u_{1, 2} = \frac{- ( - 3 ) \pm 41}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 41}{2 \cdot 4}, u_{2} = \frac{- ( - 3 ) - 41}{2 \cdot 4}

u = \frac{- ( - 3 ) + 41}{2 \cdot 4} : \frac{3 + 41}{8}

\frac{- ( - 3 ) + 41}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{3 + 41}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{3 + 41}{8}

u = \frac{- ( - 3 ) - 41}{2 \cdot 4} : \frac{3 - 41}{8}

\frac{- ( - 3 ) - 41}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{3 - 41}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{3 - 41}{8}

二次方程组的解是:

u = \frac{3 + 41}{8}, u = \frac{3 - 41}{8}

u = sin (x)

代回

sin (x) = \frac{3 + 41}{8}, sin (x) = \frac{3 - 41}{8}

sin (x) = \frac{3 + 41}{8}, sin (x) = \frac{3 - 41}{8}

sin (x) = \frac{3 + 41}{8} :

无解

sin (x) = \frac{3 + 41}{8}

- 1 \leq sin (x) \leq 1

无解

sin (x) = \frac{3 - 41}{8} : x = arcsin (\frac{3 - 41}{8}) + 2 πn, x = π + arcsin (- \frac{3 - 41}{8}) + 2 πn

sin (x) = \frac{3 - 41}{8}

使用反三角函数性质

sin (x) = \frac{3 - 41}{8}

sin (x) = \frac{3 - 41}{8}

的通解

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

x = arcsin (\frac{3 - 41}{8}) + 2 πn, x = π + arcsin (- \frac{3 - 41}{8}) + 2 πn

x = arcsin (\frac{3 - 41}{8}) + 2 πn, x = π + arcsin (- \frac{3 - 41}{8}) + 2 πn

合并所有解

x = arcsin (\frac{3 - 41}{8}) + 2 πn, x = π + arcsin (- \frac{3 - 41}{8}) + 2 πn

以小数形式表示解

x = - 0.43939 \dots + 2 πn, x = π + 0.43939 \dots + 2 πn

作图

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