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受欢迎的三角函数 >

cos^2(x)-cos(x)=0.75

解答

cos^{2} (x) - cos (x) = 0.75

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) - cos (x) = 0.75

用替代法求解

cos^{2} (x) - cos (x) = 0.75

令

: cos (x) = u

u^{2} - u = 0.75

u^{2} - u = 0.75 : u = \frac{3}{2}, u = - \frac{1}{2}

u^{2} - u = 0.75

在两边乘以

100

u^{2} - u = 0.75

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

u^{2} \cdot 100 - u \cdot 100 = 0.75 \cdot 100

整理后得

u^{2} \cdot 100 - u \cdot 100 = 75

u^{2} \cdot 100 - u \cdot 100 = 75

将

75

para o lado esquerdo

u^{2} \cdot 100 - u \cdot 100 = 75

两边减去

75

u^{2} \cdot 100 - u \cdot 100 - 75 = 75 - 75

化简

u^{2} \cdot 100 - u \cdot 100 - 75 = 0

u^{2} \cdot 100 - u \cdot 100 - 75 = 0

改写成标准形式

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

100 u^{2} - 100 u - 75 = 0

使用求根公式求解

100 u^{2} - 100 u - 75 = 0

二次方程求根公式:

若

a = 100, b = - 100, c = - 75

u_{1, 2} = \frac{- ( - 100 ) \pm ( - 100 ) ^{2} - 4 \cdot 100 ( - 75 )}{2 \cdot 100}

u_{1, 2} = \frac{- ( - 100 ) \pm ( - 100 ) ^{2} - 4 \cdot 100 ( - 75 )}{2 \cdot 100}

(- 100)^{2} - 4 \cdot 100 (- 75) = 200

(- 100)^{2} - 4 \cdot 100 (- 75)

使用法则

- (- a) = a

= (- 100)^{2} + 4 \cdot 100 \cdot 75

使用指数法则:

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

若

n

是偶数

(- 100)^{2} = 10 0^{2}

= 10 0^{2} + 4 \cdot 100 \cdot 75

数字相乘:

4 \cdot 100 \cdot 75 = 30000

= 10 0^{2} + 30000

10 0^{2} = 10000

= 10000 + 30000

数字相加:

10000 + 30000 = 40000

= 40000

因式分解数字:

40000 = 20 0^{2}

= 20 0^{2}

使用根式运算法则:

n a^{n} = a

20 0^{2} = 200

= 200

u_{1, 2} = \frac{- ( - 100 ) \pm 200}{2 \cdot 100}

将解分隔开

u_{1} = \frac{- ( - 100 ) + 200}{2 \cdot 100}, u_{2} = \frac{- ( - 100 ) - 200}{2 \cdot 100}

u = \frac{- ( - 100 ) + 200}{2 \cdot 100} : \frac{3}{2}

\frac{- ( - 100 ) + 200}{2 \cdot 100}

使用法则

- (- a) = a

= \frac{100 + 200}{2 \cdot 100}

数字相加:

100 + 200 = 300

= \frac{300}{2 \cdot 100}

数字相乘:

2 \cdot 100 = 200

= \frac{300}{200}

约分:

100

= \frac{3}{2}

u = \frac{- ( - 100 ) - 200}{2 \cdot 100} : - \frac{1}{2}

\frac{- ( - 100 ) - 200}{2 \cdot 100}

使用法则

- (- a) = a

= \frac{100 - 200}{2 \cdot 100}

数字相减:

100 - 200 = - 100

= \frac{- 100}{2 \cdot 100}

数字相乘:

2 \cdot 100 = 200

= \frac{- 100}{200}

使用分式法则:

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

= - \frac{100}{200}

约分:

100

= - \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

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

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

cos (x) = \frac{3}{2} :

无解

cos (x) = \frac{3}{2}

- 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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