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受欢迎的三角函数 >

cos^4(x)= 3/8+1/2 cos^2(x)+1/8 cos^4(x)

解答

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

解答

x = 2 πn, x = π + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

用替代法求解

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

令

: cos (x) = u

u^{4} = \frac{3}{8} + \frac{1}{2} u^{2} + \frac{1}{8} u^{4}

u^{4} = \frac{3}{8} + \frac{1}{2} u^{2} + \frac{1}{8} u^{4} : u = i \frac{3}{7}, u = - i \frac{3}{7}, u = 1, u = - 1

u^{4} = \frac{3}{8} + \frac{1}{2} u^{2} + \frac{1}{8} u^{4}

找到

8, 2

的最小公倍数:

8

8, 2

最小公倍数 (LCM)

8

质因数分解:

2 \cdot 2 \cdot 2

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

质因数分解:

2

2

2

是质数，因此无法因数分解

= 2

将每个因子乘以它在

8

或

2

中出现的最多次数

= 2 \cdot 2 \cdot 2

数字相乘:

2 \cdot 2 \cdot 2 = 8

= 8

乘以最小公倍数=

8

u^{4} \cdot 8 = \frac{3}{8} \cdot 8 + \frac{1}{2} u^{2} \cdot 8 + \frac{1}{8} u^{4} \cdot 8

化简

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

交换两边

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

将

8 u^{4}

para o lado esquerdo

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

两边减去

8 u^{4}

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

化简

3 + 4 u^{2} - 7 u^{4} = 0

3 + 4 u^{2} - 7 u^{4} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

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

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

- 7 v^{2} + 4 v + 3 = 0

解

- 7 v^{2} + 4 v + 3 = 0 : v = - \frac{3}{7}, v = 1

- 7 v^{2} + 4 v + 3 = 0

使用求根公式求解

- 7 v^{2} + 4 v + 3 = 0

二次方程求根公式:

若

a = - 7, b = 4, c = 3

v_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 7 ) \cdot 3}{2 ( - 7 )}

v_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 7 ) \cdot 3}{2 ( - 7 )}

4^{2} - 4 (- 7) \cdot 3 = 10

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

使用法则

- (- a) = a

= 4^{2} + 4 \cdot 7 \cdot 3

数字相乘:

4 \cdot 7 \cdot 3 = 84

= 4^{2} + 84

4^{2} = 16

= 16 + 84

数字相加:

16 + 84 = 100

= 100

因式分解数字:

100 = 1 0^{2}

= 1 0^{2}

使用根式运算法则:

n a^{n} = a

1 0^{2} = 10

= 10

v_{1, 2} = \frac{- 4 \pm 10}{2 ( - 7 )}

将解分隔开

v_{1} = \frac{- 4 + 10}{2 ( - 7 )}, v_{2} = \frac{- 4 - 10}{2 ( - 7 )}

v = \frac{- 4 + 10}{2 ( - 7 )} : - \frac{3}{7}

\frac{- 4 + 10}{2 ( - 7 )}

去除括号:

(- a) = - a

= \frac{- 4 + 10}{- 2 \cdot 7}

数字相加/相减:

- 4 + 10 = 6

= \frac{6}{- 2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{6}{- 14}

使用分式法则:

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

= - \frac{6}{14}

约分:

2

= - \frac{3}{7}

v = \frac{- 4 - 10}{2 ( - 7 )} : 1

\frac{- 4 - 10}{2 ( - 7 )}

去除括号:

(- a) = - a

= \frac{- 4 - 10}{- 2 \cdot 7}

数字相减:

- 4 - 10 = - 14

= \frac{- 14}{- 2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{- 14}{- 14}

使用分式法则:

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

= \frac{14}{14}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

v = - \frac{3}{7}, v = 1

v = - \frac{3}{7}, v = 1

代回

v = u^{2}

，求解

u

解

u^{2} = - \frac{3}{7} : u = i \frac{3}{7}, u = - i \frac{3}{7}

u^{2} = - \frac{3}{7}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = - \frac{3}{7}, u = - - \frac{3}{7}

化简

- \frac{3}{7} : i \frac{3}{7}

- \frac{3}{7}

使用根式运算法则:

- a = - 1 a

- \frac{3}{7} = - 1 \frac{3}{7}

= - 1 \frac{3}{7}

使用虚数运算法则:

- 1 = i

= i \frac{3}{7}

化简

- - \frac{3}{7} : - i \frac{3}{7}

- - \frac{3}{7}

化简

- \frac{3}{7} : i \frac{3}{7}

- \frac{3}{7}

使用根式运算法则:

- a = - 1 a

- \frac{3}{7} = - 1 \frac{3}{7}

= - 1 \frac{3}{7}

使用虚数运算法则:

- 1 = i

= i \frac{3}{7}

= - i \frac{3}{7}

u = i \frac{3}{7}, u = - i \frac{3}{7}

解

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

使用法则

1 = 1

= 1

- 1 = - 1

- 1

使用法则

1 = 1

= - 1

u = 1, u = - 1

解为

u = i \frac{3}{7}, u = - i \frac{3}{7}, u = 1, u = - 1

u = cos (x)

代回

cos (x) = i \frac{3}{7}, cos (x) = - i \frac{3}{7}, cos (x) = 1, cos (x) = - 1

cos (x) = i \frac{3}{7}, cos (x) = - i \frac{3}{7}, cos (x) = 1, cos (x) = - 1

cos (x) = i \frac{3}{7} :

无解

cos (x) = i \frac{3}{7}

无解

cos (x) = - i \frac{3}{7} :

无解

cos (x) = - i \frac{3}{7}

无解

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) = - 1 : x = π + 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 = π + 2 πn

x = π + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

sin(2x)=5cos(x)

sin (2 x) = 5 cos (x)

sin (a) = 0.4848

sin^2(x)=2cos^4(x)

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

sin^3(x)+cos^3(x)=(1-1)/(2sin^2(x))

sin^{3} (x) + cos^{3} (x) = \frac{1 - 1}{2 sin ^{2} ( x )}

solvefor i,xsin^2(x)=cos^2(x)

so l v e f or i, x sin^{2} (x) = cos^{2} (x)