zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

3cos(2θ)=9cos(θ)-6

解答

3 cos (2 θ) = 9 cos (θ) - 6

解答

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

+1

度数

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

求解步骤

3 cos (2 θ) = 9 cos (θ) - 6

两边减去

9 cos (θ) - 6

3 cos (2 θ) - 9 cos (θ) + 6 = 0

使用三角恒等式改写

6 + 3 cos (2 θ) - 9 cos (θ)

使用倍角公式:

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

= 6 + 3 (2 cos^{2} (θ) - 1) - 9 cos (θ)

化简

6 + 3 (2 cos^{2} (θ) - 1) - 9 cos (θ) : 6 cos^{2} (θ) - 9 cos (θ) + 3

6 + 3 (2 cos^{2} (θ) - 1) - 9 cos (θ)

乘开

3 (2 cos^{2} (θ) - 1) : 6 cos^{2} (θ) - 3

3 (2 cos^{2} (θ) - 1)

使用分配律:

a (b - c) = ab - a c

a = 3, b = 2 cos^{2} (θ), c = 1

= 3 \cdot 2 cos^{2} (θ) - 3 \cdot 1

化简

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

3 \cdot 2 cos^{2} (θ) - 3 \cdot 1

数字相乘:

3 \cdot 2 = 6

= 6 cos^{2} (θ) - 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

= 6 cos^{2} (θ) - 3

= 6 cos^{2} (θ) - 3

= 6 + 6 cos^{2} (θ) - 3 - 9 cos (θ)

化简

6 + 6 cos^{2} (θ) - 3 - 9 cos (θ) : 6 cos^{2} (θ) - 9 cos (θ) + 3

6 + 6 cos^{2} (θ) - 3 - 9 cos (θ)

对同类项分组

= 6 cos^{2} (θ) - 9 cos (θ) + 6 - 3

数字相加/相减:

6 - 3 = 3

= 6 cos^{2} (θ) - 9 cos (θ) + 3

= 6 cos^{2} (θ) - 9 cos (θ) + 3

= 6 cos^{2} (θ) - 9 cos (θ) + 3

3 + 6 cos^{2} (θ) - 9 cos (θ) = 0

用替代法求解

3 + 6 cos^{2} (θ) - 9 cos (θ) = 0

令

: cos (θ) = u

3 + 6 u^{2} - 9 u = 0

3 + 6 u^{2} - 9 u = 0 : u = 1, u = \frac{1}{2}

3 + 6 u^{2} - 9 u = 0

改写成标准形式

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

6 u^{2} - 9 u + 3 = 0

使用求根公式求解

6 u^{2} - 9 u + 3 = 0

二次方程求根公式:

若

a = 6, b = - 9, c = 3

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

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

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

使用指数法则:

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

若

n

是偶数

(- 9)^{2} = 9^{2}

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

数字相乘:

4 \cdot 6 \cdot 3 = 72

= 9^{2} - 72

9^{2} = 81

= 81 - 72

数字相减:

81 - 72 = 9

= 9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相加:

9 + 3 = 12

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

数字相乘:

2 \cdot 6 = 12

= \frac{12}{12}

使用法则

\frac{a}{a} = 1

= 1

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

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

使用法则

- (- a) = a

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

数字相减:

9 - 3 = 6

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

数字相乘:

2 \cdot 6 = 12

= \frac{6}{12}

约分:

6

= \frac{1}{2}

二次方程组的解是:

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

u = cos (θ)

代回

cos (θ) = 1, cos (θ) = \frac{1}{2}

cos (θ) = 1, cos (θ) = \frac{1}{2}

cos (θ) = 1 : θ = 2 πn

cos (θ) = 1

cos (θ) = 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}

θ = 0 + 2 πn

θ = 0 + 2 πn

解

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn

cos (θ) = \frac{1}{2} : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

cos (θ) = \frac{1}{2}

cos (θ) = \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}

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

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

合并所有解

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

作图

流行的例子

3cosh(x)+5sinh(x)=-3

3 cosh (x) + 5 sinh (x) = - 3

6cos^2(x)-5cos(x)+1=0

6 cos^{2} (x) - 5 cos (x) + 1 = 0

6 cos (x) = 5

6cos(2x)=6cos^2(x)-5

6 cos (2 x) = 6 cos^{2} (x) - 5

(-4cos(x)-6sin(x))^2-20sin^2(x)=16

(- 4 cos (x) - 6 sin (x))^{2} - 20 sin^{2} (x) = 16