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受欢迎的三角函数 >

4cos(x)=4cos(2x)

解答

4 cos (x) = 4 cos (2 x)

解答

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

+1

度数

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

求解步骤

4 cos (x) = 4 cos (2 x)

两边减去

4 cos (2 x)

4 cos (x) - 4 cos (2 x) = 0

使用三角恒等式改写

- 4 cos (2 x) + 4 cos (x)

使用倍角公式:

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

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

- (- 1 + 2 cos^{2} (x)) \cdot 4 + 4 cos (x) = 0

用替代法求解

- (- 1 + 2 cos^{2} (x)) \cdot 4 + 4 cos (x) = 0

令

: cos (x) = u

- (- 1 + 2 u^{2}) \cdot 4 + 4 u = 0

- (- 1 + 2 u^{2}) \cdot 4 + 4 u = 0 : u = - \frac{1}{2}, u = 1

- (- 1 + 2 u^{2}) \cdot 4 + 4 u = 0

展开

- (- 1 + 2 u^{2}) \cdot 4 + 4 u : 4 - 8 u^{2} + 4 u

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

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

乘开

- 4 (- 1 + 2 u^{2}) : 4 - 8 u^{2}

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

使用分配律:

a (b + c) = ab + a c

a = - 4, b = - 1, c = 2 u^{2}

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

使用加减运算法则

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

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

化简

4 \cdot 1 - 4 \cdot 2 u^{2} : 4 - 8 u^{2}

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

数字相乘:

4 \cdot 1 = 4

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

数字相乘:

4 \cdot 2 = 8

= 4 - 8 u^{2}

= 4 - 8 u^{2}

= 4 - 8 u^{2} + 4 u

4 - 8 u^{2} + 4 u = 0

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 8, b = 4, c = 4

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 8 ) \cdot 4}{2 ( - 8 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 8 ) \cdot 4}{2 ( - 8 )}

4^{2} - 4 (- 8) \cdot 4 = 12

4^{2} - 4 (- 8) \cdot 4

使用法则

- (- a) = a

= 4^{2} + 4 \cdot 8 \cdot 4

数字相乘:

4 \cdot 8 \cdot 4 = 128

= 4^{2} + 128

4^{2} = 16

= 16 + 128

数字相加:

16 + 128 = 144

= 144

因式分解数字:

144 = 1 2^{2}

= 1 2^{2}

使用根式运算法则:

n a^{n} = a

1 2^{2} = 12

= 12

u_{1, 2} = \frac{- 4 \pm 12}{2 ( - 8 )}

将解分隔开

u_{1} = \frac{- 4 + 12}{2 ( - 8 )}, u_{2} = \frac{- 4 - 12}{2 ( - 8 )}

u = \frac{- 4 + 12}{2 ( - 8 )} : - \frac{1}{2}

\frac{- 4 + 12}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 4 + 12}{- 2 \cdot 8}

数字相加/相减:

- 4 + 12 = 8

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

数字相乘:

2 \cdot 8 = 16

= \frac{8}{- 16}

使用分式法则:

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

= - \frac{8}{16}

约分:

8

= - \frac{1}{2}

u = \frac{- 4 - 12}{2 ( - 8 )} : 1

\frac{- 4 - 12}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 4 - 12}{- 2 \cdot 8}

数字相减:

- 4 - 12 = - 16

= \frac{- 16}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 16}{- 16}

使用分式法则:

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

= \frac{16}{16}

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{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) = 1 : x = 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 = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

合并所有解

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

作图

流行的例子

2 sin (\frac{π}{3} - x) = 1

sin(2x)sin(4x)=cos(2x)cos(4x)

sin (2 x) sin (4 x) = cos (2 x) cos (4 x)

solvefor x,2sin(5x)-1=0

so l v e f or x, 2 sin (5 x) - 1 = 0

2sin((pix)/2)+1=0

2 sin (\frac{π x}{2}) + 1 = 0

sin (x) = 0.819