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受欢迎的三角函数 >

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

解答

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

解答

x = 0.80515 \dots + 2 πn, x = 2 π - 0.80515 \dots + 2 πn

+1

度数

x = 46.13190 \dots^{\circ} + 36 0^{\circ} n, x = 313.86809 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 2 + 2 cos (2 x) + 3 cos (x)

使用倍角公式:

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

= - 2 + 2 (2 cos^{2} (x) - 1) + 3 cos (x)

化简

- 2 + 2 (2 cos^{2} (x) - 1) + 3 cos (x) : 4 cos^{2} (x) + 3 cos (x) - 4

- 2 + 2 (2 cos^{2} (x) - 1) + 3 cos (x)

乘开

2 (2 cos^{2} (x) - 1) : 4 cos^{2} (x) - 2

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

使用分配律:

a (b - c) = ab - a c

a = 2, b = 2 cos^{2} (x), c = 1

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

化简

2 \cdot 2 cos^{2} (x) - 2 \cdot 1 : 4 cos^{2} (x) - 2

2 \cdot 2 cos^{2} (x) - 2 \cdot 1

数字相乘:

2 \cdot 2 = 4

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

数字相乘:

2 \cdot 1 = 2

= 4 cos^{2} (x) - 2

= 4 cos^{2} (x) - 2

= - 2 + 4 cos^{2} (x) - 2 + 3 cos (x)

化简

- 2 + 4 cos^{2} (x) - 2 + 3 cos (x) : 4 cos^{2} (x) + 3 cos (x) - 4

- 2 + 4 cos^{2} (x) - 2 + 3 cos (x)

对同类项分组

= 4 cos^{2} (x) + 3 cos (x) - 2 - 2

数字相减:

- 2 - 2 = - 4

= 4 cos^{2} (x) + 3 cos (x) - 4

= 4 cos^{2} (x) + 3 cos (x) - 4

= 4 cos^{2} (x) + 3 cos (x) - 4

- 4 + 3 cos (x) + 4 cos^{2} (x) = 0

用替代法求解

- 4 + 3 cos (x) + 4 cos^{2} (x) = 0

令

: cos (x) = u

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

- 4 + 3 u + 4 u^{2} = 0 : u = \frac{- 3 + 73}{8}, u = \frac{- 3 - 73}{8}

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

3^{2} - 4 \cdot 4 (- 4) = 73

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 4 \cdot 4 = 64

= 3^{2} + 64

3^{2} = 9

= 9 + 64

数字相加:

9 + 64 = 73

= 73

u_{1, 2} = \frac{- 3 \pm 73}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 3 + 73}{2 \cdot 4}, u_{2} = \frac{- 3 - 73}{2 \cdot 4}

u = \frac{- 3 + 73}{2 \cdot 4} : \frac{- 3 + 73}{8}

\frac{- 3 + 73}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 3 + 73}{8}

u = \frac{- 3 - 73}{2 \cdot 4} : \frac{- 3 - 73}{8}

\frac{- 3 - 73}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 3 - 73}{8}

二次方程组的解是:

u = \frac{- 3 + 73}{8}, u = \frac{- 3 - 73}{8}

u = cos (x)

代回

cos (x) = \frac{- 3 + 73}{8}, cos (x) = \frac{- 3 - 73}{8}

cos (x) = \frac{- 3 + 73}{8}, cos (x) = \frac{- 3 - 73}{8}

cos (x) = \frac{- 3 + 73}{8} : x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

cos (x) = \frac{- 3 + 73}{8}

使用反三角函数性质

cos (x) = \frac{- 3 + 73}{8}

cos (x) = \frac{- 3 + 73}{8}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

cos (x) = \frac{- 3 - 73}{8} :

无解

cos (x) = \frac{- 3 - 73}{8}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

以小数形式表示解

x = 0.80515 \dots + 2 πn, x = 2 π - 0.80515 \dots + 2 πn

作图

流行的例子

sqrt(6)=-sqrt(8)cos(3x)

6 = - 8 cos (3 x)

sin(θ)=0.12,8<= θ<360

sin (θ) = 0.12, 8 \leq θ < 36 0^{\circ}

2cos(7x)-sqrt(3)=0

2 cos (7 x) - 3 = 0

sec(x)-1=0,0<= x<= 2pi

sec (x) - 1 = 0, 0 \leq x \leq 2 π

(sin(95)}{8.95}=\frac{sin(x))/5

\frac{sin ( 9 5 ^{\circ} )}{8.95} = \frac{sin ( x )}{5}