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受欢迎的三角函数 >

sin(3x)=3sin(x)cos(x)

解答

sin (3 x) = 3 sin (x) cos (x)

解答

x = 2 πn, x = π + 2 πn, x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = - 104.47751 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (3 x) = 3 sin (x) cos (x)

两边减去

3 sin (x) cos (x)

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

使用三角恒等式改写

sin (3 x) - 3 cos (x) sin (x)

sin (3 x) = 3 sin (x) - 4 sin^{3} (x)

sin (3 x)

使用三角恒等式改写

sin (3 x)

改写为

= sin (2 x + x)

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (2 x) cos (x) + cos (2 x) sin (x)

使用倍角公式:

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

= cos (2 x) sin (x) + cos (x) 2 sin (x) cos (x)

化简

cos (2 x) sin (x) + cos (x) \cdot 2 sin (x) cos (x) : sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

cos (2 x) sin (x) + cos (x) 2 sin (x) cos (x)

cos (x) \cdot 2 sin (x) cos (x) = 2 cos^{2} (x) sin (x)

cos (x) 2 sin (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

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

数字相加:

1 + 1 = 2

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

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

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

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

使用倍角公式:

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

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

使用毕达哥拉斯恒等式:

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

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

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

乘开

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

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x)

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

乘开

sin (x) (1 - 2 sin^{2} (x)) : sin (x) - 2 sin^{3} (x)

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

使用分配律:

a (b - c) = ab - a c

a = sin (x), b = 1, c = 2 sin^{2} (x)

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

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

化简

1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : sin (x) - 2 sin^{3} (x)

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

1 \cdot sin (x) = sin (x)

1 sin (x)

乘以:

1 \cdot sin (x) = sin (x)

= sin (x)

2 sin^{2} (x) sin (x) = 2 sin^{3} (x)

2 sin^{2} (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= 2 sin^{2 + 1} (x)

数字相加:

2 + 1 = 3

= 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x)

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

乘开

2 sin (x) (1 - sin^{2} (x)) : 2 sin (x) - 2 sin^{3} (x)

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

使用分配律:

a (b - c) = ab - a c

a = 2 sin (x), b = 1, c = sin^{2} (x)

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

= 2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

化简

2 \cdot 1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : 2 sin (x) - 2 sin^{3} (x)

2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 sin (x)

数字相乘:

2 \cdot 1 = 2

= 2 sin (x)

2 sin^{2} (x) sin (x) = 2 sin^{3} (x)

2 sin^{2} (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= 2 sin^{2 + 1} (x)

数字相加:

2 + 1 = 3

= 2 sin^{3} (x)

= 2 sin (x) - 2 sin^{3} (x)

= 2 sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

化简

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

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

对同类项分组

= - 2 sin^{3} (x) - 2 sin^{3} (x) + sin (x) + 2 sin (x)

同类项相加:

- 2 sin^{3} (x) - 2 sin^{3} (x) = - 4 sin^{3} (x)

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

同类项相加:

sin (x) + 2 sin (x) = 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= 3 sin (x) - 4 sin^{3} (x) - 3 cos (x) sin (x)

3 sin (x) - 4 sin^{3} (x) - 3 cos (x) sin (x) = 0

分解

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

3 sin (x) - 4 sin^{3} (x) - 3 cos (x) sin (x)

使用指数法则:

a^{b + c} = a^{b} a^{c}

sin^{3} (x) = sin (x) sin^{2} (x)

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

因式分解出通项

sin (x)

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

sin (x) (3 - 4 sin^{2} (x) - 3 cos (x)) = 0

分别求解每个部分

sin (x) = 0 or 3 - 4 sin^{2} (x) - 3 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

3 - 4 sin^{2} (x) - 3 cos (x) = 0 : x = 2 πn, x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

3 - 4 sin^{2} (x) - 3 cos (x) = 0

使用三角恒等式改写

3 - 3 cos (x) - 4 sin^{2} (x)

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = - 4, b = 1, c = cos^{2} (x)

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

使用加减运算法则

- (- a) = a

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

数字相乘:

4 \cdot 1 = 4

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

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

化简

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

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

对同类项分组

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

数字相加/相减:

3 - 4 = - 1

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

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

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

- 1 - 3 cos (x) + 4 cos^{2} (x) = 0

用替代法求解

- 1 - 3 cos (x) + 4 cos^{2} (x) = 0

令

: cos (x) = u

- 1 - 3 u + 4 u^{2} = 0

- 1 - 3 u + 4 u^{2} = 0 : u = 1, u = - \frac{1}{4}

- 1 - 3 u + 4 u^{2} = 0

改写成标准形式

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

4 u^{2} - 3 u - 1 = 0

使用求根公式求解

4 u^{2} - 3 u - 1 = 0

二次方程求根公式:

若

a = 4, b = - 3, c = - 1

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 4 ( - 1 )}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 4 ( - 1 )}{2 \cdot 4}

(- 3)^{2} - 4 \cdot 4 (- 1) = 5

(- 3)^{2} - 4 \cdot 4 (- 1)

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 4 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 4 \cdot 1

数字相乘:

4 \cdot 4 \cdot 1 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 5}{2 \cdot 4}, u_{2} = \frac{- ( - 3 ) - 5}{2 \cdot 4}

u = \frac{- ( - 3 ) + 5}{2 \cdot 4} : 1

\frac{- ( - 3 ) + 5}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{3 + 5}{2 \cdot 4}

数字相加:

3 + 5 = 8

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

数字相乘:

2 \cdot 4 = 8

= \frac{8}{8}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- ( - 3 ) - 5}{2 \cdot 4} : - \frac{1}{4}

\frac{- ( - 3 ) - 5}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{3 - 5}{2 \cdot 4}

数字相减:

3 - 5 = - 2

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 2}{8}

使用分式法则:

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

= - \frac{2}{8}

约分:

2

= - \frac{1}{4}

二次方程组的解是:

u = 1, u = - \frac{1}{4}

u = cos (x)

代回

cos (x) = 1, cos (x) = - \frac{1}{4}

cos (x) = 1, cos (x) = - \frac{1}{4}

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

cos (x) = - \frac{1}{4} : x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

cos (x) = - \frac{1}{4}

使用反三角函数性质

cos (x) = - \frac{1}{4}

cos (x) = - \frac{1}{4}

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

合并所有解

x = 2 πn, x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

作图

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