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受欢迎的三角函数 >

cos^4(x)-3cos^2(x)=4

解答

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

解答

x \in R 无解

求解步骤

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

用替代法求解

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

令

: cos (x) = u

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

u^{4} - 3 u^{2} = 4 : u = 2, u = - 2, u = i, u = - i

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

将

4

para o lado esquerdo

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

两边减去

4

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

化简

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

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

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

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

解

v^{2} - 3 v - 4 = 0 : v = 4, v = - 1

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = - 3, c = - 4

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

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

(- 3)^{2} - 4 \cdot 1 \cdot (- 4) = 5

(- 3)^{2} - 4 \cdot 1 \cdot (- 4)

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 1 \cdot 4

使用指数法则:

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

若

n

是偶数

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

= 3^{2} + 4 \cdot 1 \cdot 4

数字相乘:

4 \cdot 1 \cdot 4 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

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

将解分隔开

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

v = \frac{- ( - 3 ) + 5}{2 \cdot 1} : 4

\frac{- ( - 3 ) + 5}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{3 + 5}{2 \cdot 1}

数字相加:

3 + 5 = 8

= \frac{8}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8}{2}

数字相除:

\frac{8}{2} = 4

= 4

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

\frac{- ( - 3 ) - 5}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{3 - 5}{2 \cdot 1}

数字相减:

3 - 5 = - 2

= \frac{- 2}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 2}{2}

使用分式法则:

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

= - \frac{2}{2}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

v = 4, v = - 1

v = 4, v = - 1

代回

v = u^{2}

，求解

u

解

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

对于

x^{2} = f (a)

解为

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

u = 4, u = - 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

解

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

u^{2} = - 1

对于

x^{2} = f (a)

解为

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

u = - 1, u = - - 1

化简

- 1 : i

- 1

使用虚数运算法则:

- 1 = i

= i

化简

- - 1 : - i

- - 1

使用虚数运算法则:

- 1 = i

= - i

u = i, u = - i

解为

u = 2, u = - 2, u = i, u = - i

u = cos (x)

代回

cos (x) = 2, cos (x) = - 2, cos (x) = i, cos (x) = - i

cos (x) = 2, cos (x) = - 2, cos (x) = i, cos (x) = - i

cos (x) = 2 :

无解

cos (x) = 2

- 1 \leq cos (x) \leq 1

无解

cos (x) = - 2 :

无解

cos (x) = - 2

- 1 \leq cos (x) \leq 1

无解

cos (x) = i :

无解

cos (x) = i

无解

cos (x) = - i :

无解

cos (x) = - i

无解

合并所有解

x \in R 无解

作图

流行的例子

10cos(θ)=-5cos(θ)

10 cos (θ) = - 5 cos (θ)

tan (2 θ) = 1.33

5*cos(x)+4-2*sin(x)=0

5 \cdot cos (x) + 4 - 2 \cdot sin (x) = 0

sin (x) = - 0.54

tan (2 θ) = 1.82