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受欢迎的三角函数 >

sin^2(x)-3cos(x)-4=0

解答

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

解答

x \in R 无解

求解步骤

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

使用三角恒等式改写

- 4 + sin^{2} (x) - 3 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 + 1 - cos^{2} (x) - 3 cos (x)

化简

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

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

用替代法求解

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

令

: cos (x) = u

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

- 3 - u^{2} - 3 u = 0 : u = - \frac{3}{2} - i \frac{3}{2}, u = - \frac{3}{2} + i \frac{3}{2}

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

化简

(- 3)^{2} - 4 (- 1) (- 3) : 3 i

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

使用法则

- (- a) = a

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

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 3

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 3^{2} - 12

使用虚数运算法则:

- a = i a

= i 12 - 3^{2}

- 3^{2} + 12 = 3

- 3^{2} + 12

3^{2} = 9

= - 9 + 12

数字相加/相减:

- 9 + 12 = 3

= 3

= 3 i

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

将解分隔开

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

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

\frac{- ( - 3 ) + 3 i}{2 ( - 1 )}

去除括号:

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

= \frac{3 + 3 i}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{3 + 3 i}{- 2}

使用分式法则:

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

= - \frac{3 + 3 i}{2}

将

- \frac{3 + 3 i}{2}

改写成标准复数形式:

- \frac{3}{2} - \frac{3}{2} i

- \frac{3 + 3 i}{2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{3 + 3 i}{2} = - (\frac{3}{2}) - (\frac{3 i}{2})

= - (\frac{3}{2}) - (\frac{3 i}{2})

去除括号:

(a) = a

= - \frac{3}{2} - \frac{3 i}{2}

= - \frac{3}{2} - \frac{3}{2} i

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

\frac{- ( - 3 ) - 3 i}{2 ( - 1 )}

去除括号:

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

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

数字相乘:

2 \cdot 1 = 2

= \frac{3 - 3 i}{- 2}

使用分式法则:

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

= - \frac{3 - 3 i}{2}

将

- \frac{3 - 3 i}{2}

改写成标准复数形式:

- \frac{3}{2} + \frac{3}{2} i

- \frac{3 - 3 i}{2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{3 - 3 i}{2} = - (\frac{3}{2}) - (- \frac{3 i}{2})

= - (\frac{3}{2}) - (- \frac{3 i}{2})

去除括号:

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

= - \frac{3}{2} + \frac{3 i}{2}

= - \frac{3}{2} + \frac{3}{2} i

二次方程组的解是:

u = - \frac{3}{2} - i \frac{3}{2}, u = - \frac{3}{2} + i \frac{3}{2}

u = cos (x)

代回

cos (x) = - \frac{3}{2} - i \frac{3}{2}, cos (x) = - \frac{3}{2} + i \frac{3}{2}

cos (x) = - \frac{3}{2} - i \frac{3}{2}, cos (x) = - \frac{3}{2} + i \frac{3}{2}

cos (x) = - \frac{3}{2} - i \frac{3}{2} :

无解

cos (x) = - \frac{3}{2} - i \frac{3}{2}

无解

cos (x) = - \frac{3}{2} + i \frac{3}{2} :

无解

cos (x) = - \frac{3}{2} + i \frac{3}{2}

无解

合并所有解

x \in R 无解

作图

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