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受欢迎的三角函数 >

6cos^2(x)-5cos(x)=4

解答

6 cos^{2} (x) - 5 cos (x) = 4

解答

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

6 cos^{2} (x) - 5 cos (x) = 4

用替代法求解

6 cos^{2} (x) - 5 cos (x) = 4

令

: cos (x) = u

6 u^{2} - 5 u = 4

6 u^{2} - 5 u = 4 : u = \frac{4}{3}, u = - \frac{1}{2}

6 u^{2} - 5 u = 4

将

4

para o lado esquerdo

6 u^{2} - 5 u = 4

两边减去

4

6 u^{2} - 5 u - 4 = 4 - 4

化简

6 u^{2} - 5 u - 4 = 0

6 u^{2} - 5 u - 4 = 0

使用求根公式求解

6 u^{2} - 5 u - 4 = 0

二次方程求根公式:

若

a = 6, b = - 5, c = - 4

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 6 ( - 4 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 6 ( - 4 )}{2 \cdot 6}

(- 5)^{2} - 4 \cdot 6 (- 4) = 11

(- 5)^{2} - 4 \cdot 6 (- 4)

使用法则

- (- a) = a

= (- 5)^{2} + 4 \cdot 6 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 6 \cdot 4

数字相乘:

4 \cdot 6 \cdot 4 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

数字相加:

25 + 96 = 121

= 121

因式分解数字:

121 = 1 1^{2}

= 1 1^{2}

使用根式运算法则:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 5 ) \pm 11}{2 \cdot 6}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 11}{2 \cdot 6}, u_{2} = \frac{- ( - 5 ) - 11}{2 \cdot 6}

u = \frac{- ( - 5 ) + 11}{2 \cdot 6} : \frac{4}{3}

\frac{- ( - 5 ) + 11}{2 \cdot 6}

使用法则

- (- a) = a

= \frac{5 + 11}{2 \cdot 6}

数字相加:

5 + 11 = 16

= \frac{16}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{16}{12}

约分:

4

= \frac{4}{3}

u = \frac{- ( - 5 ) - 11}{2 \cdot 6} : - \frac{1}{2}

\frac{- ( - 5 ) - 11}{2 \cdot 6}

使用法则

- (- a) = a

= \frac{5 - 11}{2 \cdot 6}

数字相减:

5 - 11 = - 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 = \frac{4}{3}, u = - \frac{1}{2}

u = cos (x)

代回

cos (x) = \frac{4}{3}, cos (x) = - \frac{1}{2}

cos (x) = \frac{4}{3}, cos (x) = - \frac{1}{2}

cos (x) = \frac{4}{3} :

无解

cos (x) = \frac{4}{3}

- 1 \leq cos (x) \leq 1

无解

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

合并所有解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

作图

流行的例子

cos (2 0^{\circ}) = sin (x)

sec (2 x) = 1

2cos^2(x)+sin^2(x)=0

2 cos^{2} (x) + sin^{2} (x) = 0

3cos(2x)-5sin(x)+1=0

3 cos (2 x) - 5 sin (x) + 1 = 0

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

sin (x + \frac{π}{2}) = \frac{1}{2}