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受欢迎的三角函数 >

2cos^2(x)-8cos(x)=2sin^2(x)-5

解答

2 cos^{2} (x) - 8 cos (x) = 2 sin^{2} (x) - 5

解答

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

2 cos^{2} (x) - 8 cos (x) = 2 sin^{2} (x) - 5

两边减去

2 sin^{2} (x) - 5

2 cos^{2} (x) - 8 cos (x) - 2 sin^{2} (x) + 5 = 0

使用三角恒等式改写

5 + 2 cos^{2} (x) - 2 sin^{2} (x) - 8 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)

= 5 + 2 cos^{2} (x) - 2 (1 - cos^{2} (x)) - 8 cos (x)

化简

5 + 2 cos^{2} (x) - 2 (1 - cos^{2} (x)) - 8 cos (x) : 4 cos^{2} (x) - 8 cos (x) + 3

5 + 2 cos^{2} (x) - 2 (1 - cos^{2} (x)) - 8 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)

使用加减运算法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

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

= 5 + 2 cos^{2} (x) - 2 + 2 cos^{2} (x) - 8 cos (x)

化简

5 + 2 cos^{2} (x) - 2 + 2 cos^{2} (x) - 8 cos (x) : 4 cos^{2} (x) - 8 cos (x) + 3

5 + 2 cos^{2} (x) - 2 + 2 cos^{2} (x) - 8 cos (x)

对同类项分组

= 2 cos^{2} (x) + 2 cos^{2} (x) - 8 cos (x) + 5 - 2

同类项相加:

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

= 4 cos^{2} (x) - 8 cos (x) + 5 - 2

数字相加/相减:

5 - 2 = 3

= 4 cos^{2} (x) - 8 cos (x) + 3

= 4 cos^{2} (x) - 8 cos (x) + 3

= 4 cos^{2} (x) - 8 cos (x) + 3

3 + 4 cos^{2} (x) - 8 cos (x) = 0

用替代法求解

3 + 4 cos^{2} (x) - 8 cos (x) = 0

令

: cos (x) = u

3 + 4 u^{2} - 8 u = 0

3 + 4 u^{2} - 8 u = 0 : u = \frac{3}{2}, u = \frac{1}{2}

3 + 4 u^{2} - 8 u = 0

改写成标准形式

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

4 u^{2} - 8 u + 3 = 0

使用求根公式求解

4 u^{2} - 8 u + 3 = 0

二次方程求根公式:

若

a = 4, b = - 8, c = 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

(- 8)^{2} - 4 \cdot 4 \cdot 3 = 4

(- 8)^{2} - 4 \cdot 4 \cdot 3

使用指数法则:

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

若

n

是偶数

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 4 \cdot 3

数字相乘:

4 \cdot 4 \cdot 3 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

数字相减:

64 - 48 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- ( - 8 ) \pm 4}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 8 ) + 4}{2 \cdot 4}, u_{2} = \frac{- ( - 8 ) - 4}{2 \cdot 4}

u = \frac{- ( - 8 ) + 4}{2 \cdot 4} : \frac{3}{2}

\frac{- ( - 8 ) + 4}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{8 + 4}{2 \cdot 4}

数字相加:

8 + 4 = 12

= \frac{12}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{12}{8}

约分:

4

= \frac{3}{2}

u = \frac{- ( - 8 ) - 4}{2 \cdot 4} : \frac{1}{2}

\frac{- ( - 8 ) - 4}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{8 - 4}{2 \cdot 4}

数字相减:

8 - 4 = 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{4}{8}

约分:

4

= \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{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

合并所有解

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

流行的例子

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sin^2(x)+5sin(x)+5=0

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

cos (x) = - 0.814

3cos^2(x)-8cos(x)+4=sin^2(x)

3 cos^{2} (x) - 8 cos (x) + 4 = sin^{2} (x)

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cos (x - 4 5^{\circ}) = \frac{- 1}{2}