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受欢迎的三角函数 >

6cos(2x)=-6cos(x)

解答

6 cos (2 x) = - 6 cos (x)

解答

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

6 cos (2 x) = - 6 cos (x)

两边减去

- 6 cos (x)

6 cos (2 x) + 6 cos (x) = 0

使用三角恒等式改写

6 cos (2 x) + 6 cos (x)

使用倍角公式:

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

= 6 (2 cos^{2} (x) - 1) + 6 cos (x)

(- 1 + 2 cos^{2} (x)) \cdot 6 + 6 cos (x) = 0

用替代法求解

(- 1 + 2 cos^{2} (x)) \cdot 6 + 6 cos (x) = 0

令

: cos (x) = u

(- 1 + 2 u^{2}) \cdot 6 + 6 u = 0

(- 1 + 2 u^{2}) \cdot 6 + 6 u = 0 : u = \frac{1}{2}, u = - 1

(- 1 + 2 u^{2}) \cdot 6 + 6 u = 0

展开

(- 1 + 2 u^{2}) \cdot 6 + 6 u : - 6 + 12 u^{2} + 6 u

(- 1 + 2 u^{2}) \cdot 6 + 6 u

= 6 (- 1 + 2 u^{2}) + 6 u

乘开

6 (- 1 + 2 u^{2}) : - 6 + 12 u^{2}

6 (- 1 + 2 u^{2})

使用分配律:

a (b + c) = ab + a c

a = 6, b = - 1, c = 2 u^{2}

= 6 (- 1) + 6 \cdot 2 u^{2}

使用加减运算法则

+ (- a) = - a

= - 6 \cdot 1 + 6 \cdot 2 u^{2}

化简

- 6 \cdot 1 + 6 \cdot 2 u^{2} : - 6 + 12 u^{2}

- 6 \cdot 1 + 6 \cdot 2 u^{2}

数字相乘:

6 \cdot 1 = 6

= - 6 + 6 \cdot 2 u^{2}

数字相乘:

6 \cdot 2 = 12

= - 6 + 12 u^{2}

= - 6 + 12 u^{2}

= - 6 + 12 u^{2} + 6 u

- 6 + 12 u^{2} + 6 u = 0

改写成标准形式

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

12 u^{2} + 6 u - 6 = 0

使用求根公式求解

12 u^{2} + 6 u - 6 = 0

二次方程求根公式:

若

a = 12, b = 6, c = - 6

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 12 ( - 6 )}{2 \cdot 12}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 12 ( - 6 )}{2 \cdot 12}

6^{2} - 4 \cdot 12 (- 6) = 18

6^{2} - 4 \cdot 12 (- 6)

使用法则

- (- a) = a

= 6^{2} + 4 \cdot 12 \cdot 6

数字相乘:

4 \cdot 12 \cdot 6 = 288

= 6^{2} + 288

6^{2} = 36

= 36 + 288

数字相加:

36 + 288 = 324

= 324

因式分解数字:

324 = 1 8^{2}

= 1 8^{2}

使用根式运算法则:

n a^{n} = a

1 8^{2} = 18

= 18

u_{1, 2} = \frac{- 6 \pm 18}{2 \cdot 12}

将解分隔开

u_{1} = \frac{- 6 + 18}{2 \cdot 12}, u_{2} = \frac{- 6 - 18}{2 \cdot 12}

u = \frac{- 6 + 18}{2 \cdot 12} : \frac{1}{2}

\frac{- 6 + 18}{2 \cdot 12}

数字相加/相减:

- 6 + 18 = 12

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

数字相乘:

2 \cdot 12 = 24

= \frac{12}{24}

约分:

12

= \frac{1}{2}

u = \frac{- 6 - 18}{2 \cdot 12} : - 1

\frac{- 6 - 18}{2 \cdot 12}

数字相减:

- 6 - 18 = - 24

= \frac{- 24}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{- 24}{24}

使用分式法则:

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

= - \frac{24}{24}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

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

u = cos (x)

代回

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

作图

流行的例子

sec(2x)csc(2x)=2csc(2x),0<= x<= 2pi

sec (2 x) csc (2 x) = 2 csc (2 x), 0 \leq x \leq 2 π

tan^2(x)-3tan(x)-10=0

tan^{2} (x) - 3 tan (x) - 10 = 0

6 tan^{2} (x) - 18 = 0

11sin^2(θ)-5sin(θ)=4

11 sin^{2} (θ) - 5 sin (θ) = 4

5cos(x)cot(x)=1

5 cos (x) cot (x) = 1