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受欢迎的三角函数 >

2cos^2(x)+9cos(x)+4=0

解答

2 cos^{2} (x) + 9 cos (x) + 4 = 0

解答

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

+1

度数

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

求解步骤

2 cos^{2} (x) + 9 cos (x) + 4 = 0

用替代法求解

2 cos^{2} (x) + 9 cos (x) + 4 = 0

令

: cos (x) = u

2 u^{2} + 9 u + 4 = 0

2 u^{2} + 9 u + 4 = 0 : u = - \frac{1}{2}, u = - 4

2 u^{2} + 9 u + 4 = 0

使用求根公式求解

2 u^{2} + 9 u + 4 = 0

二次方程求根公式:

若

a = 2, b = 9, c = 4

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 2 \cdot 4}{2 \cdot 2}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 2 \cdot 4}{2 \cdot 2}

9^{2} - 4 \cdot 2 \cdot 4 = 7

9^{2} - 4 \cdot 2 \cdot 4

数字相乘:

4 \cdot 2 \cdot 4 = 32

= 9^{2} - 32

9^{2} = 81

= 81 - 32

数字相减:

81 - 32 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 9 \pm 7}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 9 + 7}{2 \cdot 2}, u_{2} = \frac{- 9 - 7}{2 \cdot 2}

u = \frac{- 9 + 7}{2 \cdot 2} : - \frac{1}{2}

\frac{- 9 + 7}{2 \cdot 2}

数字相加/相减:

- 9 + 7 = - 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

u = \frac{- 9 - 7}{2 \cdot 2} : - 4

\frac{- 9 - 7}{2 \cdot 2}

数字相减:

- 9 - 7 = - 16

= \frac{- 16}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 16}{4}

使用分式法则:

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

= - \frac{16}{4}

数字相除:

\frac{16}{4} = 4

= - 4

二次方程组的解是:

u = - \frac{1}{2}, u = - 4

u = cos (x)

代回

cos (x) = - \frac{1}{2}, cos (x) = - 4

cos (x) = - \frac{1}{2}, cos (x) = - 4

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

cos (x) = - 4 :

无解

cos (x) = - 4

- 1 \leq cos (x) \leq 1

无解

合并所有解

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

作图

流行的例子

cos(6x)=sin(x-1)

cos (6 x) = sin (x - 1)

tan (θ) = \frac{72}{65}

cos(x)+cos(x/2)+1=0

cos (x) + cos (\frac{x}{2}) + 1 = 0

sin(θ)=(1.5)/2

sin (θ) = \frac{1.5}{2}

-cos(θ)=cos(θ)

- cos (θ) = cos (θ)