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受欢迎的三角函数 >

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

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

2

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

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

化简

- 2 + 1 - sin^{2} (x) - 4 sin^{2} (x) : - 5 sin^{2} (x) - 1

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

同类项相加:

- sin^{2} (x) - 4 sin^{2} (x) = - 5 sin^{2} (x)

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

数字相加/相减:

- 2 + 1 = - 1

= - 5 sin^{2} (x) - 1

= - 5 sin^{2} (x) - 1

- 1 - 5 sin^{2} (x) = 0

用替代法求解

- 1 - 5 sin^{2} (x) = 0

令

: sin (x) = u

- 1 - 5 u^{2} = 0

- 1 - 5 u^{2} = 0 : u = i \frac{1}{5}, u = - i \frac{1}{5}

- 1 - 5 u^{2} = 0

将

1

到右边

- 1 - 5 u^{2} = 0

两边加上

1

- 1 - 5 u^{2} + 1 = 0 + 1

化简

- 5 u^{2} = 1

- 5 u^{2} = 1

两边除以

- 5

- 5 u^{2} = 1

两边除以

- 5

\frac{- 5 u ^{2}}{- 5} = \frac{1}{- 5}

化简

u^{2} = - \frac{1}{5}

u^{2} = - \frac{1}{5}

对于

x^{2} = f (a)

解为

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

u = - \frac{1}{5}, u = - - \frac{1}{5}

化简

- \frac{1}{5} : i \frac{1}{5}

- \frac{1}{5}

使用根式运算法则:

- a = - 1 a

- \frac{1}{5} = - 1 \frac{1}{5}

= - 1 \frac{1}{5}

使用虚数运算法则:

- 1 = i

= i \frac{1}{5}

化简

- - \frac{1}{5} : - i \frac{1}{5}

- - \frac{1}{5}

化简

- \frac{1}{5} : i \frac{1}{5}

- \frac{1}{5}

使用根式运算法则:

- a = - 1 a

- \frac{1}{5} = - 1 \frac{1}{5}

= - 1 \frac{1}{5}

使用虚数运算法则:

- 1 = i

= i \frac{1}{5}

= - i \frac{1}{5}

u = i \frac{1}{5}, u = - i \frac{1}{5}

u = sin (x)

代回

sin (x) = i \frac{1}{5}, sin (x) = - i \frac{1}{5}

sin (x) = i \frac{1}{5}, sin (x) = - i \frac{1}{5}

sin (x) = i \frac{1}{5} :

无解

sin (x) = i \frac{1}{5}

无解

sin (x) = - i \frac{1}{5} :

无解

sin (x) = - i \frac{1}{5}

无解

合并所有解

x \in R 无解

作图

流行的例子

sin (x) = 0.53

2cos(2θ)=sqrt(2)

2 cos (2 θ) = 2

2 + sin (x) = 0

sin (x) = 0.45

sin (x) = 0.42