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受欢迎的三角函数 >

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

解答

6 sin (\frac{x}{2}) + 6 cos (x) = 0

解答

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = π + 4 πn

+1

度数

x = 42 0^{\circ} + 72 0^{\circ} n, x = 66 0^{\circ} + 72 0^{\circ} n, x = 18 0^{\circ} + 72 0^{\circ} n

求解步骤

6 sin (\frac{x}{2}) + 6 cos (x) = 0

令

: u = \frac{x}{2}

6 sin (u) + 6 cos (2 u) = 0

使用三角恒等式改写

6 cos (2 u) + 6 sin (u)

使用倍角公式:

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

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

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

用替代法求解

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

令

: sin (u) = 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 \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 ( - 12 ) \cdot 6}{2 ( - 12 )}

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

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

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

将解分隔开

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

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

\frac{- 6 + 18}{2 ( - 12 )}

去除括号:

(- a) = - a

= \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}

使用分式法则:

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

= - \frac{12}{24}

约分:

12

= - \frac{1}{2}

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

\frac{- 6 - 18}{2 ( - 12 )}

去除括号:

(- a) = - a

= \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 = sin (u)

代回

sin (u) = - \frac{1}{2}, sin (u) = 1

sin (u) = - \frac{1}{2}, sin (u) = 1

sin (u) = - \frac{1}{2} : u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

sin (u) = - \frac{1}{2}

sin (u) = - \frac{1}{2}

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

sin (u) = 1 : u = \frac{π}{2} + 2 πn

sin (u) = 1

sin (u) = 1

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = \frac{π}{2} + 2 πn

u = \frac{π}{2} + 2 πn

合并所有解

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn, u = \frac{π}{2} + 2 πn

u = \frac{x}{2}

代回

\frac{x}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{7 π}{6} + 2 πn

在两边乘以

2

\frac{x}{2} = \frac{7 π}{6} + 2 πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

化简

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 π 2}{6}

数字相乘:

7 \cdot 2 = 14

= \frac{14 π}{6}

约分:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn

在两边乘以

2

\frac{x}{2} = \frac{11 π}{6} + 2 πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

化简

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{11 π 2}{6}

数字相乘:

11 \cdot 2 = 22

= \frac{22 π}{6}

约分:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn : x = π + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

约分:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = π + 4 πn

作图

流行的例子

cos (x) = 0.925

(sin(A))/9 =(sin(108))/6

\frac{sin ( A )}{9} = \frac{sin ( 10 8 ^{\circ} )}{6}

sin (3 x - 6^{\circ}) = \frac{1}{2}

sin (θ) = \frac{12}{17}

sin(x/2)+(sqrt(3))/2 =0

sin (\frac{x}{2}) + \frac{3}{2} = 0