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受欢迎的三角函数 >

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

解答

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

解答

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

+1

度数

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cos (x) = u

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

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

4^{2} - 4 \cdot 3 (- 4) = 8

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数字相加:

16 + 48 = 64

= 64

因式分解数字:

64 = 8^{2}

= 8^{2}

使用根式运算法则:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- 4 \pm 8}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- 4 + 8}{2 \cdot 3}, u_{2} = \frac{- 4 - 8}{2 \cdot 3}

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

\frac{- 4 + 8}{2 \cdot 3}

数字相加/相减:

- 4 + 8 = 4

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

数字相乘:

2 \cdot 3 = 6

= \frac{4}{6}

约分:

2

= \frac{2}{3}

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

\frac{- 4 - 8}{2 \cdot 3}

数字相减:

- 4 - 8 = - 12

= \frac{- 12}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 12}{6}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{12}{6}

数字相除:

\frac{12}{6} = 2

= - 2

二次方程组的解是:

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

u = cos (x)

代回

cos (x) = \frac{2}{3}, cos (x) = - 2

cos (x) = \frac{2}{3}, cos (x) = - 2

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

cos (x) = \frac{2}{3}

使用反三角函数性质

cos (x) = \frac{2}{3}

cos (x) = \frac{2}{3}

的通解

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

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

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

cos (x) = - 2 :

无解

cos (x) = - 2

- 1 \leq cos (x) \leq 1

无解

合并所有解

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

以小数形式表示解

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

作图

流行的例子

sin(θ)=-27/190

sin (θ) = - \frac{27}{190}

4sin(θ)=1+sin(θ)

4 sin (θ) = 1 + sin (θ)

tan(θ)-sqrt(3)=0,0<= θ<= 2pi

tan (θ) - 3 = 0, 0 \leq θ \leq 2 π

1/32*1200^2*sin(2θ)=3000

\frac{1}{32} \cdot 120 0^{2} \cdot sin (2 θ) = 3000

1-cos^2(a)=sin^2(2a)

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