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受欢迎的三角函数 >

5sin^2(x)+10sin(x)+2=0

解答

5 sin^{2} (x) + 10 sin (x) + 2 = 0

解答

x = - 0.22735 \dots + 2 πn, x = π + 0.22735 \dots + 2 πn

+1

度数

x = - 13.02659 \dots^{\circ} + 36 0^{\circ} n, x = 193.02659 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 sin^{2} (x) + 10 sin (x) + 2 = 0

用替代法求解

5 sin^{2} (x) + 10 sin (x) + 2 = 0

令

: sin (x) = u

5 u^{2} + 10 u + 2 = 0

5 u^{2} + 10 u + 2 = 0 : u = \frac{- 5 + 15}{5}, u = - \frac{5 + 15}{5}

5 u^{2} + 10 u + 2 = 0

使用求根公式求解

5 u^{2} + 10 u + 2 = 0

二次方程求根公式:

若

a = 5, b = 10, c = 2

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 5 \cdot 2}{2 \cdot 5}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 5 \cdot 2}{2 \cdot 5}

1 0^{2} - 4 \cdot 5 \cdot 2 = 215

1 0^{2} - 4 \cdot 5 \cdot 2

数字相乘:

4 \cdot 5 \cdot 2 = 40

= 1 0^{2} - 40

1 0^{2} = 100

= 100 - 40

数字相减:

100 - 40 = 60

= 60

60

质因数分解:

2^{2} \cdot 3 \cdot 5

60

60

除以

260 = 30 \cdot 2

= 2 \cdot 30

30

除以

230 = 15 \cdot 2

= 2 \cdot 2 \cdot 15

15

除以

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 5

2, 3, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 5

= 2^{2} \cdot 3 \cdot 5

= 2^{2} \cdot 3 \cdot 5

使用根式运算法则:

n ab = n a n b

= 2^{2} 3 \cdot 5

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 3 \cdot 5

整理后得

= 215

u_{1, 2} = \frac{- 10 \pm 2 15}{2 \cdot 5}

将解分隔开

u_{1} = \frac{- 10 + 2 15}{2 \cdot 5}, u_{2} = \frac{- 10 - 2 15}{2 \cdot 5}

u = \frac{- 10 + 2 15}{2 \cdot 5} : \frac{- 5 + 15}{5}

\frac{- 10 + 2 15}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 10 + 2 15}{10}

分解

- 10 + 215 : 2 (- 5 + 15)

- 10 + 215

改写为

= - 2 \cdot 5 + 215

因式分解出通项

2

= 2 (- 5 + 15)

= \frac{2 ( - 5 + 15 )}{10}

约分:

2

= \frac{- 5 + 15}{5}

u = \frac{- 10 - 2 15}{2 \cdot 5} : - \frac{5 + 15}{5}

\frac{- 10 - 2 15}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 10 - 2 15}{10}

分解

- 10 - 215 : - 2 (5 + 15)

- 10 - 215

改写为

= - 2 \cdot 5 - 215

因式分解出通项

2

= - 2 (5 + 15)

= - \frac{2 ( 5 + 15 )}{10}

约分:

2

= - \frac{5 + 15}{5}

二次方程组的解是:

u = \frac{- 5 + 15}{5}, u = - \frac{5 + 15}{5}

u = sin (x)

代回

sin (x) = \frac{- 5 + 15}{5}, sin (x) = - \frac{5 + 15}{5}

sin (x) = \frac{- 5 + 15}{5}, sin (x) = - \frac{5 + 15}{5}

sin (x) = \frac{- 5 + 15}{5} : x = arcsin (\frac{- 5 + 15}{5}) + 2 πn, x = π + arcsin (- \frac{- 5 + 15}{5}) + 2 πn

sin (x) = \frac{- 5 + 15}{5}

使用反三角函数性质

sin (x) = \frac{- 5 + 15}{5}

sin (x) = \frac{- 5 + 15}{5}

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (\frac{- 5 + 15}{5}) + 2 πn, x = π + arcsin (- \frac{- 5 + 15}{5}) + 2 πn

x = arcsin (\frac{- 5 + 15}{5}) + 2 πn, x = π + arcsin (- \frac{- 5 + 15}{5}) + 2 πn

sin (x) = - \frac{5 + 15}{5} :

无解

sin (x) = - \frac{5 + 15}{5}

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (\frac{- 5 + 15}{5}) + 2 πn, x = π + arcsin (- \frac{- 5 + 15}{5}) + 2 πn

以小数形式表示解

x = - 0.22735 \dots + 2 πn, x = π + 0.22735 \dots + 2 πn

作图

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