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受欢迎的三角函数 >

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

解答

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

解答

x = π + 2 πn

+1

度数

x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

3 + cos (2 x) + 4 cos (x)

使用倍角公式:

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

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

化简

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

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

对同类项分组

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

数字相加/相减:

3 - 1 = 2

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

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

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

用替代法求解

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

令

: cos (x) = u

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

2 + 2 u^{2} + 4 u = 0 : u = - 1

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 2, b = 4, c = 2

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

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

4^{2} - 4 \cdot 2 \cdot 2 = 0

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

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

数字相减:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- 4 \pm 0}{2 \cdot 2}

u = \frac{- 4}{2 \cdot 2}

\frac{- 4}{2 \cdot 2} = - 1

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{4}

使用分式法则:

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

= - \frac{4}{4}

使用法则

\frac{a}{a} = 1

= - 1

u = - 1

二次方程组的解是:

u = - 1

u = cos (x)

代回

cos (x) = - 1

cos (x) = - 1

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

合并所有解

x = π + 2 πn

作图

流行的例子

2cos(θ)-1=sec(θ)

2 cos (θ) - 1 = sec (θ)

3cos(x)-6sin(x)cos(x)=0

3 cos (x) - 6 sin (x) cos (x) = 0

sin (30 x) = - 0.6

sec^2(x)-2tan(x)-4=0

sec^{2} (x) - 2 tan (x) - 4 = 0

cos(40-x)=cos(x)

cos (4 0^{\circ} - x) = cos (x)