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受欢迎的三角函数 >

3sin^2(x)-10sin(x)-8=0

解答

3 sin^{2} (x) - 10 sin (x) - 8 = 0

解答

x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn

+1

度数

x = - 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 221.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 sin^{2} (x) - 10 sin (x) - 8 = 0

用替代法求解

3 sin^{2} (x) - 10 sin (x) - 8 = 0

令

: sin (x) = u

3 u^{2} - 10 u - 8 = 0

3 u^{2} - 10 u - 8 = 0 : u = 4, u = - \frac{2}{3}

3 u^{2} - 10 u - 8 = 0

使用求根公式求解

3 u^{2} - 10 u - 8 = 0

二次方程求根公式:

若

a = 3, b = - 10, c = - 8

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 3 ( - 8 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 3 ( - 8 )}{2 \cdot 3}

(- 10)^{2} - 4 \cdot 3 (- 8) = 14

(- 10)^{2} - 4 \cdot 3 (- 8)

使用法则

- (- a) = a

= (- 10)^{2} + 4 \cdot 3 \cdot 8

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 10)^{2} = 1 0^{2}

= 1 0^{2} + 4 \cdot 3 \cdot 8

数字相乘:

4 \cdot 3 \cdot 8 = 96

= 1 0^{2} + 96

1 0^{2} = 100

= 100 + 96

数字相加:

100 + 96 = 196

= 196

因式分解数字:

196 = 1 4^{2}

= 1 4^{2}

使用根式运算法则:

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- ( - 10 ) \pm 14}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 10 ) + 14}{2 \cdot 3}, u_{2} = \frac{- ( - 10 ) - 14}{2 \cdot 3}

u = \frac{- ( - 10 ) + 14}{2 \cdot 3} : 4

\frac{- ( - 10 ) + 14}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{10 + 14}{2 \cdot 3}

数字相加:

10 + 14 = 24

= \frac{24}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{24}{6}

数字相除:

\frac{24}{6} = 4

= 4

u = \frac{- ( - 10 ) - 14}{2 \cdot 3} : - \frac{2}{3}

\frac{- ( - 10 ) - 14}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{10 - 14}{2 \cdot 3}

数字相减:

10 - 14 = - 4

= \frac{- 4}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 4}{6}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{6}

约分:

2

= - \frac{2}{3}

二次方程组的解是:

u = 4, u = - \frac{2}{3}

u = sin (x)

代回

sin (x) = 4, sin (x) = - \frac{2}{3}

sin (x) = 4, sin (x) = - \frac{2}{3}

sin (x) = 4 :

无解

sin (x) = 4

- 1 \leq sin (x) \leq 1

无解

sin (x) = - \frac{2}{3} : x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn

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

使用反三角函数性质

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

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

的通解

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

x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn

x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn

合并所有解

x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn

以小数形式表示解

x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn

作图

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