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受欢迎的三角函数 >

9sin^2(x)-6sin(x)+1=0

解答

9 sin^{2} (x) - 6 sin (x) + 1 = 0

解答

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

+1

度数

x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n

求解步骤

9 sin^{2} (x) - 6 sin (x) + 1 = 0

用替代法求解

9 sin^{2} (x) - 6 sin (x) + 1 = 0

令

: sin (x) = u

9 u^{2} - 6 u + 1 = 0

9 u^{2} - 6 u + 1 = 0 : u = \frac{1}{3}

9 u^{2} - 6 u + 1 = 0

使用求根公式求解

9 u^{2} - 6 u + 1 = 0

二次方程求根公式:

若

a = 9, b = - 6, c = 1

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 9 \cdot 1}{2 \cdot 9}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 9 \cdot 1}{2 \cdot 9}

(- 6)^{2} - 4 \cdot 9 \cdot 1 = 0

(- 6)^{2} - 4 \cdot 9 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 9 \cdot 1

数字相乘:

4 \cdot 9 \cdot 1 = 36

= 6^{2} - 36

6^{2} = 36

= 36 - 36

数字相减:

36 - 36 = 0

= 0

u_{1, 2} = \frac{- ( - 6 ) \pm 0}{2 \cdot 9}

u = \frac{- ( - 6 )}{2 \cdot 9}

\frac{- ( - 6 )}{2 \cdot 9} = \frac{1}{3}

\frac{- ( - 6 )}{2 \cdot 9}

使用法则

- (- a) = a

= \frac{6}{2 \cdot 9}

数字相乘:

2 \cdot 9 = 18

= \frac{6}{18}

约分:

6

= \frac{1}{3}

u = \frac{1}{3}

二次方程组的解是:

u = \frac{1}{3}

u = sin (x)

代回

sin (x) = \frac{1}{3}

sin (x) = \frac{1}{3}

sin (x) = \frac{1}{3} : x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

sin (x) = \frac{1}{3}

使用反三角函数性质

sin (x) = \frac{1}{3}

sin (x) = \frac{1}{3}

的通解

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

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

合并所有解

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

以小数形式表示解

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

作图

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