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受欢迎的三角函数 >

-5cos(x)=-2sin^2(x)+4

解答

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

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

两边减去

- 2 sin^{2} (x) + 4

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

使用三角恒等式改写

- 4 + 2 sin^{2} (x) - 5 cos (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

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

= - 4 + 2 (1 - cos^{2} (x)) - 5 cos (x)

化简

- 4 + 2 (1 - cos^{2} (x)) - 5 cos (x) : - 2 cos^{2} (x) - 5 cos (x) - 2

- 4 + 2 (1 - cos^{2} (x)) - 5 cos (x)

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

2 \cdot 1 = 2

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

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

数字相加/相减:

- 4 + 2 = - 2

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

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

- 2 - 2 cos^{2} (x) - 5 cos (x) = 0

用替代法求解

- 2 - 2 cos^{2} (x) - 5 cos (x) = 0

令

: cos (x) = u

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

- 2 - 2 u^{2} - 5 u = 0 : u = - 2, u = - \frac{1}{2}

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

(- 5)^{2} - 4 (- 2) (- 2) = 3

(- 5)^{2} - 4 (- 2) (- 2)

使用法则

- (- a) = a

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

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 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

数字相减:

25 - 16 = 9

= 9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

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

将解分隔开

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

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

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

去除括号:

(- a) = - a, - (- a) = a

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

数字相加:

5 + 3 = 8

= \frac{8}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{8}{- 4}

使用分式法则:

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

= - \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= - 2

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

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

去除括号:

(- a) = - a, - (- a) = a

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

数字相减:

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

u = cos (x)

代回

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

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

cos (x) = - 2 :

无解

cos (x) = - 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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