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受欢迎的三角函数 >

12cos^2(x)+2cos(x)-2=0

解答

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

解答

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度数

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cos (x) = u

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

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 12, b = 2, c = - 2

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 12 ( - 2 )}{2 \cdot 12}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 12 ( - 2 )}{2 \cdot 12}

2^{2} - 4 \cdot 12 (- 2) = 10

2^{2} - 4 \cdot 12 (- 2)

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 12 \cdot 2

数字相乘:

4 \cdot 12 \cdot 2 = 96

= 2^{2} + 96

2^{2} = 4

= 4 + 96

数字相加:

4 + 96 = 100

= 100

因式分解数字:

100 = 1 0^{2}

= 1 0^{2}

使用根式运算法则:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 2 \pm 10}{2 \cdot 12}

将解分隔开

u_{1} = \frac{- 2 + 10}{2 \cdot 12}, u_{2} = \frac{- 2 - 10}{2 \cdot 12}

u = \frac{- 2 + 10}{2 \cdot 12} : \frac{1}{3}

\frac{- 2 + 10}{2 \cdot 12}

数字相加/相减:

- 2 + 10 = 8

= \frac{8}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{8}{24}

约分:

8

= \frac{1}{3}

u = \frac{- 2 - 10}{2 \cdot 12} : - \frac{1}{2}

\frac{- 2 - 10}{2 \cdot 12}

数字相减:

- 2 - 10 = - 12

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

数字相乘:

2 \cdot 12 = 24

= \frac{- 12}{24}

使用分式法则:

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

= - \frac{12}{24}

约分:

12

= - \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

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

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

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

cos (x) = \frac{1}{3}

使用反三角函数性质

cos (x) = \frac{1}{3}

cos (x) = \frac{1}{3}

的通解

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

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

合并所有解

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

以小数形式表示解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

作图

流行的例子

sin (θ) = - 0.94

sin(θ)=0,0<= θ<= 2pi

sin (θ) = 0, 0 \leq θ \leq 2 π

sin (θ) = - 0.71

2 cos (4 x) = 0

tan(2θ)=(-1)/1

tan (2 θ) = \frac{- 1}{1}