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受欢迎的三角函数 >

sin^2(x)+7sin^2(x)+5cos^2(x)+3=0

解答

sin^{2} (x) + 7 sin^{2} (x) + 5 cos^{2} (x) + 3 = 0

解答

x \in R 无解

求解步骤

sin^{2} (x) + 7 sin^{2} (x) + 5 cos^{2} (x) + 3 = 0

使用三角恒等式改写

3 + sin^{2} (x) + 5 cos^{2} (x) + 7 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)

= 3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) + 7 sin^{2} (x)

化简

3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) + 7 sin^{2} (x) : 3 sin^{2} (x) + 8

3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) + 7 sin^{2} (x)

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = 5, b = 1, c = sin^{2} (x)

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

数字相乘:

5 \cdot 1 = 5

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

= 3 + sin^{2} (x) + 5 - 5 sin^{2} (x) + 7 sin^{2} (x)

化简

3 + sin^{2} (x) + 5 - 5 sin^{2} (x) + 7 sin^{2} (x) : 3 sin^{2} (x) + 8

3 + sin^{2} (x) + 5 - 5 sin^{2} (x) + 7 sin^{2} (x)

对同类项分组

= sin^{2} (x) - 5 sin^{2} (x) + 7 sin^{2} (x) + 3 + 5

同类项相加:

sin^{2} (x) - 5 sin^{2} (x) + 7 sin^{2} (x) = 3 sin^{2} (x)

= 3 sin^{2} (x) + 3 + 5

数字相加:

3 + 5 = 8

= 3 sin^{2} (x) + 8

= 3 sin^{2} (x) + 8

= 3 sin^{2} (x) + 8

8 + 3 sin^{2} (x) = 0

用替代法求解

8 + 3 sin^{2} (x) = 0

令

: sin (x) = u

8 + 3 u^{2} = 0

8 + 3 u^{2} = 0 : u = i \frac{2 6}{3}, u = - i \frac{2 6}{3}

8 + 3 u^{2} = 0

将

8

到右边

8 + 3 u^{2} = 0

两边减去

8

8 + 3 u^{2} - 8 = 0 - 8

化简

3 u^{2} = - 8

3 u^{2} = - 8

两边除以

3

3 u^{2} = - 8

两边除以

3

\frac{3 u ^{2}}{3} = \frac{- 8}{3}

化简

u^{2} = - \frac{8}{3}

u^{2} = - \frac{8}{3}

对于

x^{2} = f (a)

解为

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

u = - \frac{8}{3}, u = - - \frac{8}{3}

化简

- \frac{8}{3} : i \frac{2 6}{3}

- \frac{8}{3}

使用根式运算法则:

- a = - 1 a

- \frac{8}{3} = - 1 \frac{8}{3}

= - 1 \frac{8}{3}

使用虚数运算法则:

- 1 = i

= i \frac{8}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

\frac{8}{3} = \frac{8}{3}

= i \frac{8}{3}

8 = 22

8

8

质因数分解:

2^{3}

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22

= i \frac{2 2}{3}

\frac{2 2}{3} = \frac{2 6}{3}

\frac{2 2}{3}

乘以共轭根式

\frac{3}{3}

= \frac{2 2 3}{3 3}

223 = 26

223

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 26

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{2 6}{3}

= i \frac{2 6}{3}

将

i \frac{2 6}{3}

改写成标准复数形式:

\frac{2 6}{3} i

i \frac{2 6}{3}

\frac{2 6}{3} = \frac{2 2}{3}

\frac{2 6}{3}

分解

6 : 23

因式分解

6 = 2 \cdot 3

= 2 \cdot 3

使用根式运算法则:

n ab = n a n b

= 23

= \frac{2 2 3}{3}

消掉

\frac{2 2 3}{3} : \frac{2 2}{3}

\frac{2 2 3}{3}

使用根式运算法则:

n a = a^{\frac{1}{n}}

3 = 3^{\frac{1}{2}}

= \frac{2 2 \cdot 3 ^{\frac{1}{2}}}{3}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{1}{2}}}{3 ^{1}} = \frac{1}{3 ^{1 - \frac{1}{2}}}

= \frac{2 2}{3 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{2 2}{3 ^{\frac{1}{2}}}

使用根式运算法则:

a^{\frac{1}{n}} = n a

3^{\frac{1}{2}} = 3

= \frac{2 2}{3}

= \frac{2 2}{3}

= i \frac{2 2}{3}

分式相乘:

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

= \frac{2 2 i}{3}

\frac{2 2}{3} = \frac{2 6}{3}

\frac{2 2}{3}

乘以共轭根式

\frac{3}{3}

= \frac{2 2 3}{3 3}

223 = 26

223

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 26

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{2 6}{3}

= \frac{2 6}{3} i

= \frac{2 6}{3} i

化简

- - \frac{8}{3} : - i \frac{2 6}{3}

- - \frac{8}{3}

化简

- \frac{8}{3} : i \frac{2 2}{3}

- \frac{8}{3}

使用根式运算法则:

- a = - 1 a

- \frac{8}{3} = - 1 \frac{8}{3}

= - 1 \frac{8}{3}

使用虚数运算法则:

- 1 = i

= i \frac{8}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

\frac{8}{3} = \frac{8}{3}

= i \frac{8}{3}

8 = 22

8

8

质因数分解:

2^{3}

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22

= i \frac{2 2}{3}

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

\frac{2 2}{3} = \frac{2 6}{3}

\frac{2 2}{3}

乘以共轭根式

\frac{3}{3}

= \frac{2 2 3}{3 3}

223 = 26

223

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 26

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{2 6}{3}

= - \frac{2 6}{3} i

u = i \frac{2 6}{3}, u = - i \frac{2 6}{3}

u = sin (x)

代回

sin (x) = i \frac{2 6}{3}, sin (x) = - i \frac{2 6}{3}

sin (x) = i \frac{2 6}{3}, sin (x) = - i \frac{2 6}{3}

sin (x) = i \frac{2 6}{3} :

无解

sin (x) = i \frac{2 6}{3}

无解

sin (x) = - i \frac{2 6}{3} :

无解

sin (x) = - i \frac{2 6}{3}

无解

合并所有解

x \in R 无解

作图

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