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受欢迎的三角函数 >

3cos(2x)=3sin(x)

解答

3 cos (2 x) = 3 sin (x)

解答

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = 27 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

3 cos (2 x) = 3 sin (x)

两边减去

3 sin (x)

3 cos (2 x) - 3 sin (x) = 0

使用三角恒等式改写

3 cos (2 x) - 3 sin (x)

使用倍角公式:

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

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

(1 - 2 sin^{2} (x)) \cdot 3 - 3 sin (x) = 0

用替代法求解

(1 - 2 sin^{2} (x)) \cdot 3 - 3 sin (x) = 0

令

: sin (x) = u

(1 - 2 u^{2}) \cdot 3 - 3 u = 0

(1 - 2 u^{2}) \cdot 3 - 3 u = 0 : u = - 1, u = \frac{1}{2}

(1 - 2 u^{2}) \cdot 3 - 3 u = 0

展开

(1 - 2 u^{2}) \cdot 3 - 3 u : 3 - 6 u^{2} - 3 u

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

= 3 (1 - 2 u^{2}) - 3 u

乘开

3 (1 - 2 u^{2}) : 3 - 6 u^{2}

3 (1 - 2 u^{2})

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = 2 u^{2}

= 3 \cdot 1 - 3 \cdot 2 u^{2}

化简

3 \cdot 1 - 3 \cdot 2 u^{2} : 3 - 6 u^{2}

3 \cdot 1 - 3 \cdot 2 u^{2}

数字相乘:

3 \cdot 1 = 3

= 3 - 3 \cdot 2 u^{2}

数字相乘:

3 \cdot 2 = 6

= 3 - 6 u^{2}

= 3 - 6 u^{2}

= 3 - 6 u^{2} - 3 u

3 - 6 u^{2} - 3 u = 0

改写成标准形式

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

- 6 u^{2} - 3 u + 3 = 0

使用求根公式求解

- 6 u^{2} - 3 u + 3 = 0

二次方程求根公式:

若

a = - 6, b = - 3, c = 3

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

(- 3)^{2} - 4 (- 6) \cdot 3 = 9

(- 3)^{2} - 4 (- 6) \cdot 3

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 6 \cdot 3

使用指数法则:

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

若

n

是偶数

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

= 3^{2} + 4 \cdot 6 \cdot 3

数字相乘:

4 \cdot 6 \cdot 3 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

数字相加:

9 + 72 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 3 ) \pm 9}{2 ( - 6 )}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 9}{2 ( - 6 )}, u_{2} = \frac{- ( - 3 ) - 9}{2 ( - 6 )}

u = \frac{- ( - 3 ) + 9}{2 ( - 6 )} : - 1

\frac{- ( - 3 ) + 9}{2 ( - 6 )}

去除括号:

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

= \frac{3 + 9}{- 2 \cdot 6}

数字相加:

3 + 9 = 12

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

数字相乘:

2 \cdot 6 = 12

= \frac{12}{- 12}

使用分式法则:

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

= - \frac{12}{12}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 3 ) - 9}{2 ( - 6 )} : \frac{1}{2}

\frac{- ( - 3 ) - 9}{2 ( - 6 )}

去除括号:

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

= \frac{3 - 9}{- 2 \cdot 6}

数字相减:

3 - 9 = - 6

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

数字相乘:

2 \cdot 6 = 12

= \frac{- 6}{- 12}

使用分式法则:

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

= \frac{6}{12}

约分:

6

= \frac{1}{2}

二次方程组的解是:

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

u = sin (x)

代回

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

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

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

sin (x) = - 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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

sin (x) = \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}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

合并所有解

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

流行的例子

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4 cos (\frac{θ}{2}) + 3 = - 2 cos (θ)

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2cos(2x)+sqrt(3)=2sqrt(3)

2 cos (2 x) + 3 = 23

sin(x)cos(2x)= 1/2 (1+sin(x))

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