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受欢迎的三角函数 >

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

解答

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

解答

x = 2 πn, x = 1.98231 \dots + 2 πn, x = - 1.98231 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 113.57817 \dots^{\circ} + 36 0^{\circ} n, x = - 113.57817 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cos (x) = u

5 u^{2} - 3 u = 2

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

5 u^{2} - 3 u = 2

将

2

para o lado esquerdo

5 u^{2} - 3 u = 2

两边减去

2

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

化简

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

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

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

(- 3)^{2} = 3^{2}

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

数字相乘:

4 \cdot 5 \cdot 2 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

数字相加:

9 + 40 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 3 ) \pm 7}{2 \cdot 5}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 7}{2 \cdot 5}, u_{2} = \frac{- ( - 3 ) - 7}{2 \cdot 5}

u = \frac{- ( - 3 ) + 7}{2 \cdot 5} : 1

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

使用法则

- (- a) = a

= \frac{3 + 7}{2 \cdot 5}

数字相加:

3 + 7 = 10

= \frac{10}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{10}{10}

使用法则

\frac{a}{a} = 1

= 1

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

\frac{- ( - 3 ) - 7}{2 \cdot 5}

使用法则

- (- a) = a

= \frac{3 - 7}{2 \cdot 5}

数字相减:

3 - 7 = - 4

= \frac{- 4}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 4}{10}

使用分式法则:

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

= - \frac{4}{10}

约分:

2

= - \frac{2}{5}

二次方程组的解是:

u = 1, u = - \frac{2}{5}

u = cos (x)

代回

cos (x) = 1, cos (x) = - \frac{2}{5}

cos (x) = 1, cos (x) = - \frac{2}{5}

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{2}{5} : x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

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

使用反三角函数性质

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

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

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

合并所有解

x = 2 πn, x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

以小数形式表示解

x = 2 πn, x = 1.98231 \dots + 2 πn, x = - 1.98231 \dots + 2 πn

作图

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