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受欢迎的三角函数 >

4tan^2(x)+sec(x)=-2tan^2(x)+sec^2(x)-3

解答

4 tan^{2} (x) + sec (x) = - 2 tan^{2} (x) + sec^{2} (x) - 3

解答

x \in R 无解

求解步骤

4 tan^{2} (x) + sec (x) = - 2 tan^{2} (x) + sec^{2} (x) - 3

两边减去

- 2 tan^{2} (x) + sec^{2} (x) - 3

6 tan^{2} (x) + sec (x) - sec^{2} (x) + 3 = 0

使用三角恒等式改写

3 + sec (x) - sec^{2} (x) + 6 tan^{2} (x)

使用毕达哥拉斯恒等式:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= 3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1)

化简

3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1) : 5 sec^{2} (x) + sec (x) - 3

3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1)

乘开

6 (sec^{2} (x) - 1) : 6 sec^{2} (x) - 6

6 (sec^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = 6, b = sec^{2} (x), c = 1

= 6 sec^{2} (x) - 6 \cdot 1

数字相乘:

6 \cdot 1 = 6

= 6 sec^{2} (x) - 6

= 3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6

化简

3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6 : 5 sec^{2} (x) + sec (x) - 3

3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6

同类项相加:

- sec^{2} (x) + 6 sec^{2} (x) = 5 sec^{2} (x)

= 3 + sec (x) + 5 sec^{2} (x) - 6

对同类项分组

= sec (x) + 5 sec^{2} (x) + 3 - 6

数字相加/相减:

3 - 6 = - 3

= 5 sec^{2} (x) + sec (x) - 3

= 5 sec^{2} (x) + sec (x) - 3

= 5 sec^{2} (x) + sec (x) - 3

- 3 + sec (x) + 5 sec^{2} (x) = 0

用替代法求解

- 3 + sec (x) + 5 sec^{2} (x) = 0

令

: sec (x) = u

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

- 3 + u + 5 u^{2} = 0 : u = \frac{- 1 + 61}{10}, u = \frac{- 1 - 61}{10}

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 5, b = 1, c = - 3

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

1^{2} - 4 \cdot 5 (- 3) = 61

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 5 (- 3)

使用法则

- (- a) = a

= 1 + 4 \cdot 5 \cdot 3

数字相乘:

4 \cdot 5 \cdot 3 = 60

= 1 + 60

数字相加:

1 + 60 = 61

= 61

u_{1, 2} = \frac{- 1 \pm 61}{2 \cdot 5}

将解分隔开

u_{1} = \frac{- 1 + 61}{2 \cdot 5}, u_{2} = \frac{- 1 - 61}{2 \cdot 5}

u = \frac{- 1 + 61}{2 \cdot 5} : \frac{- 1 + 61}{10}

\frac{- 1 + 61}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 1 + 61}{10}

u = \frac{- 1 - 61}{2 \cdot 5} : \frac{- 1 - 61}{10}

\frac{- 1 - 61}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 1 - 61}{10}

二次方程组的解是:

u = \frac{- 1 + 61}{10}, u = \frac{- 1 - 61}{10}

u = sec (x)

代回

sec (x) = \frac{- 1 + 61}{10}, sec (x) = \frac{- 1 - 61}{10}

sec (x) = \frac{- 1 + 61}{10}, sec (x) = \frac{- 1 - 61}{10}

sec (x) = \frac{- 1 + 61}{10} :

无解

sec (x) = \frac{- 1 + 61}{10}

sec (x) \leq - 1 or sec (x) \geq 1

无解

sec (x) = \frac{- 1 - 61}{10} :

无解

sec (x) = \frac{- 1 - 61}{10}

sec (x) \leq - 1 or sec (x) \geq 1

无解

合并所有解

x \in R 无解

作图

流行的例子

sin (θ) = \frac{13}{14}

2 a sin (x) + a = 0

tan (θ) = (\frac{2}{3})

tan (θ) = 8.1443

4sin(x)=5cos(x)

4 sin (x) = 5 cos (x)