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受欢迎的三角函数 >

4+2cos(2x)=6cos(x)

解答

4 + 2 cos (2 x) = 6 cos (x)

解答

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

+1

度数

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

求解步骤

4 + 2 cos (2 x) = 6 cos (x)

两边减去

6 cos (x)

4 + 2 cos (2 x) - 6 cos (x) = 0

使用三角恒等式改写

4 + 2 cos (2 x) - 6 cos (x)

使用倍角公式:

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

= 4 + 2 (2 cos^{2} (x) - 1) - 6 cos (x)

化简

4 + 2 (2 cos^{2} (x) - 1) - 6 cos (x) : 4 cos^{2} (x) - 6 cos (x) + 2

4 + 2 (2 cos^{2} (x) - 1) - 6 cos (x)

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

化简

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

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

数字相乘:

2 \cdot 2 = 4

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

数字相乘:

2 \cdot 1 = 2

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

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

= 4 + 4 cos^{2} (x) - 2 - 6 cos (x)

化简

4 + 4 cos^{2} (x) - 2 - 6 cos (x) : 4 cos^{2} (x) - 6 cos (x) + 2

4 + 4 cos^{2} (x) - 2 - 6 cos (x)

对同类项分组

= 4 cos^{2} (x) - 6 cos (x) + 4 - 2

数字相加/相减:

4 - 2 = 2

= 4 cos^{2} (x) - 6 cos (x) + 2

= 4 cos^{2} (x) - 6 cos (x) + 2

= 4 cos^{2} (x) - 6 cos (x) + 2

2 + 4 cos^{2} (x) - 6 cos (x) = 0

用替代法求解

2 + 4 cos^{2} (x) - 6 cos (x) = 0

令

: cos (x) = u

2 + 4 u^{2} - 6 u = 0

2 + 4 u^{2} - 6 u = 0 : u = 1, u = \frac{1}{2}

2 + 4 u^{2} - 6 u = 0

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 4, b = - 6, c = 2

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

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

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

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

使用指数法则:

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

若

n

是偶数

(- 6)^{2} = 6^{2}

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

数字相乘:

4 \cdot 4 \cdot 2 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

数字相减:

36 - 32 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

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

将解分隔开

u_{1} = \frac{- ( - 6 ) + 2}{2 \cdot 4}, u_{2} = \frac{- ( - 6 ) - 2}{2 \cdot 4}

u = \frac{- ( - 6 ) + 2}{2 \cdot 4} : 1

\frac{- ( - 6 ) + 2}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{6 + 2}{2 \cdot 4}

数字相加:

6 + 2 = 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{- ( - 6 ) - 2}{2 \cdot 4} : \frac{1}{2}

\frac{- ( - 6 ) - 2}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{6 - 2}{2 \cdot 4}

数字相减:

6 - 2 = 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{4}{8}

约分:

4

= \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

cos (x) = 1, cos (x) = \frac{1}{2}

cos (x) = 1, cos (x) = \frac{1}{2}

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

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

合并所有解

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

作图

流行的例子

cos(x)-tan(x)cos(x)=0

cos (x) - tan (x) cos (x) = 0

cot (2 θ) = \frac{3}{4}

tan(x)=-4/(sqrt(33))

tan (x) = - \frac{4}{33}

2cot^2(x)-11csc(x)=-14

2 cot^{2} (x) - 11 csc (x) = - 14

cot(x)=0,0<= x<= 2pi

cot (x) = 0, 0 \leq x \leq 2 π