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受欢迎的三角函数 >

cos^2(x)-3/2 = 5/2 cos(x)

解答

cos^{2} (x) - \frac{3}{2} = \frac{5}{2} cos (x)

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

cos^{2} (x) - \frac{3}{2} = \frac{5}{2} cos (x)

用替代法求解

cos^{2} (x) - \frac{3}{2} = \frac{5}{2} cos (x)

令

: cos (x) = u

u^{2} - \frac{3}{2} = \frac{5}{2} u

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

u^{2} - \frac{3}{2} = \frac{5}{2} u

在两边乘以

2

u^{2} - \frac{3}{2} = \frac{5}{2} u

在两边乘以

2

u^{2} \cdot 2 - \frac{3}{2} \cdot 2 = \frac{5}{2} u \cdot 2

化简

2 u^{2} - 3 = 5 u

2 u^{2} - 3 = 5 u

将

5 u

para o lado esquerdo

2 u^{2} - 3 = 5 u

两边减去

5 u

2 u^{2} - 3 - 5 u = 5 u - 5 u

化简

2 u^{2} - 3 - 5 u = 0

2 u^{2} - 3 - 5 u = 0

改写成标准形式

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

2 u^{2} - 5 u - 3 = 0

使用求根公式求解

2 u^{2} - 5 u - 3 = 0

二次方程求根公式:

若

a = 2, b = - 5, c = - 3

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

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

(- 5)^{2} - 4 \cdot 2 (- 3) = 7

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数字相加:

25 + 24 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 7}{2 \cdot 2}, u_{2} = \frac{- ( - 5 ) - 7}{2 \cdot 2}

u = \frac{- ( - 5 ) + 7}{2 \cdot 2} : 3

\frac{- ( - 5 ) + 7}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{5 + 7}{2 \cdot 2}

数字相加:

5 + 7 = 12

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

数字相乘:

2 \cdot 2 = 4

= \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 5 ) - 7}{2 \cdot 2} : - \frac{1}{2}

\frac{- ( - 5 ) - 7}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{5 - 7}{2 \cdot 2}

数字相减:

5 - 7 = - 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

二次方程组的解是:

u = 3, u = - \frac{1}{2}

u = cos (x)

代回

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

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

cos (x) = 3 :

无解

cos (x) = 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

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