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受欢迎的三角函数 >

-2cos(x)-cos(2x)*2=0

解答

- 2 cos (x) - cos (2 x) \cdot 2 = 0

解答

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

+1

度数

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

求解步骤

- 2 cos (x) - cos (2 x) \cdot 2 = 0

使用三角恒等式改写

- 2 cos (2 x) - 2 cos (x)

使用倍角公式:

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

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

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

用替代法求解

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

令

: cos (x) = u

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

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

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

展开

- (- 1 + 2 u^{2}) \cdot 2 - 2 u : 2 - 4 u^{2} - 2 u

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

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

乘开

- 2 (- 1 + 2 u^{2}) : 2 - 4 u^{2}

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

使用分配律:

a (b + c) = ab + a c

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

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

使用加减运算法则

- (- a) = a, + (- a) = - a

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

化简

2 \cdot 1 - 2 \cdot 2 u^{2} : 2 - 4 u^{2}

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

数字相乘:

2 \cdot 1 = 2

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

数字相乘:

2 \cdot 2 = 4

= 2 - 4 u^{2}

= 2 - 4 u^{2}

= 2 - 4 u^{2} - 2 u

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 4, b = - 2, c = 2

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

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

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

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

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 4 \cdot 2

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 2)^{2} = 2^{2}

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

数字相乘:

4 \cdot 4 \cdot 2 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数字相加:

4 + 32 = 36

= 36

因式分解数字:

36 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a

6^{2} = 6

= 6

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

将解分隔开

u_{1} = \frac{- ( - 2 ) + 6}{2 ( - 4 )}, u_{2} = \frac{- ( - 2 ) - 6}{2 ( - 4 )}

u = \frac{- ( - 2 ) + 6}{2 ( - 4 )} : - 1

\frac{- ( - 2 ) + 6}{2 ( - 4 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{2 + 6}{- 2 \cdot 4}

数字相加:

2 + 6 = 8

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

数字相乘:

2 \cdot 4 = 8

= \frac{8}{- 8}

使用分式法则:

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

= - \frac{8}{8}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 2 ) - 6}{2 ( - 4 )} : \frac{1}{2}

\frac{- ( - 2 ) - 6}{2 ( - 4 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{2 - 6}{- 2 \cdot 4}

数字相减:

2 - 6 = - 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 4}{- 8}

使用分式法则:

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

= \frac{4}{8}

约分:

4

= \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

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

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

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

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

合并所有解

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

作图

流行的例子

tan^2(x)+4tan(x)+2=0

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

-9sin^2(θ)+4=5cos(θ)-1

- 9 sin^{2} (θ) + 4 = 5 cos (θ) - 1

tan (θ) = 1.25

6 sec (θ) + 6 = 0

2 sin (θ) - 2 = 0