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受欢迎的三角函数 >

7sin^2(x)+2sin(x)-2=0

解答

7 sin^{2} (x) + 2 sin (x) - 2 = 0

解答

x = 0.42292 \dots + 2 πn, x = π - 0.42292 \dots + 2 πn, x = - 0.77000 \dots + 2 πn, x = π + 0.77000 \dots + 2 πn

+1

度数

x = 24.23161 \dots^{\circ} + 36 0^{\circ} n, x = 155.76838 \dots^{\circ} + 36 0^{\circ} n, x = - 44.11816 \dots^{\circ} + 36 0^{\circ} n, x = 224.11816 \dots^{\circ} + 36 0^{\circ} n

求解步骤

7 sin^{2} (x) + 2 sin (x) - 2 = 0

用替代法求解

7 sin^{2} (x) + 2 sin (x) - 2 = 0

令

: sin (x) = u

7 u^{2} + 2 u - 2 = 0

7 u^{2} + 2 u - 2 = 0 : u = \frac{- 1 + 15}{7}, u = - \frac{1 + 15}{7}

7 u^{2} + 2 u - 2 = 0

使用求根公式求解

7 u^{2} + 2 u - 2 = 0

二次方程求根公式:

若

a = 7, b = 2, c = - 2

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 7 ( - 2 )}{2 \cdot 7}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 7 ( - 2 )}{2 \cdot 7}

2^{2} - 4 \cdot 7 (- 2) = 215

2^{2} - 4 \cdot 7 (- 2)

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 7 \cdot 2

数字相乘:

4 \cdot 7 \cdot 2 = 56

= 2^{2} + 56

2^{2} = 4

= 4 + 56

数字相加:

4 + 56 = 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{- 2 \pm 2 15}{2 \cdot 7}

将解分隔开

u_{1} = \frac{- 2 + 2 15}{2 \cdot 7}, u_{2} = \frac{- 2 - 2 15}{2 \cdot 7}

u = \frac{- 2 + 2 15}{2 \cdot 7} : \frac{- 1 + 15}{7}

\frac{- 2 + 2 15}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{- 2 + 2 15}{14}

分解

- 2 + 215 : 2 (- 1 + 15)

- 2 + 215

改写为

= - 2 \cdot 1 + 215

因式分解出通项

2

= 2 (- 1 + 15)

= \frac{2 ( - 1 + 15 )}{14}

约分:

2

= \frac{- 1 + 15}{7}

u = \frac{- 2 - 2 15}{2 \cdot 7} : - \frac{1 + 15}{7}

\frac{- 2 - 2 15}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{- 2 - 2 15}{14}

分解

- 2 - 215 : - 2 (1 + 15)

- 2 - 215

改写为

= - 2 \cdot 1 - 215

因式分解出通项

2

= - 2 (1 + 15)

= - \frac{2 ( 1 + 15 )}{14}

约分:

2

= - \frac{1 + 15}{7}

二次方程组的解是:

u = \frac{- 1 + 15}{7}, u = - \frac{1 + 15}{7}

u = sin (x)

代回

sin (x) = \frac{- 1 + 15}{7}, sin (x) = - \frac{1 + 15}{7}

sin (x) = \frac{- 1 + 15}{7}, sin (x) = - \frac{1 + 15}{7}

sin (x) = \frac{- 1 + 15}{7} : x = arcsin (\frac{- 1 + 15}{7}) + 2 πn, x = π - arcsin (\frac{- 1 + 15}{7}) + 2 πn

sin (x) = \frac{- 1 + 15}{7}

使用反三角函数性质

sin (x) = \frac{- 1 + 15}{7}

sin (x) = \frac{- 1 + 15}{7}

的通解

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

x = arcsin (\frac{- 1 + 15}{7}) + 2 πn, x = π - arcsin (\frac{- 1 + 15}{7}) + 2 πn

x = arcsin (\frac{- 1 + 15}{7}) + 2 πn, x = π - arcsin (\frac{- 1 + 15}{7}) + 2 πn

sin (x) = - \frac{1 + 15}{7} : x = arcsin (- \frac{1 + 15}{7}) + 2 πn, x = π + arcsin (\frac{1 + 15}{7}) + 2 πn

sin (x) = - \frac{1 + 15}{7}

使用反三角函数性质

sin (x) = - \frac{1 + 15}{7}

sin (x) = - \frac{1 + 15}{7}

的通解

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

x = arcsin (- \frac{1 + 15}{7}) + 2 πn, x = π + arcsin (\frac{1 + 15}{7}) + 2 πn

x = arcsin (- \frac{1 + 15}{7}) + 2 πn, x = π + arcsin (\frac{1 + 15}{7}) + 2 πn

合并所有解

x = arcsin (\frac{- 1 + 15}{7}) + 2 πn, x = π - arcsin (\frac{- 1 + 15}{7}) + 2 πn, x = arcsin (- \frac{1 + 15}{7}) + 2 πn, x = π + arcsin (\frac{1 + 15}{7}) + 2 πn

以小数形式表示解

x = 0.42292 \dots + 2 πn, x = π - 0.42292 \dots + 2 πn, x = - 0.77000 \dots + 2 πn, x = π + 0.77000 \dots + 2 πn

作图

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