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受欢迎的三角函数 >

12sin^2(x)-17sin(x)+6=0

解答

12 sin^{2} (x) - 17 sin (x) + 6 = 0

解答

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度数

x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

求解步骤

12 sin^{2} (x) - 17 sin (x) + 6 = 0

用替代法求解

12 sin^{2} (x) - 17 sin (x) + 6 = 0

令

: sin (x) = u

12 u^{2} - 17 u + 6 = 0

12 u^{2} - 17 u + 6 = 0 : u = \frac{3}{4}, u = \frac{2}{3}

12 u^{2} - 17 u + 6 = 0

使用求根公式求解

12 u^{2} - 17 u + 6 = 0

二次方程求根公式:

若

a = 12, b = - 17, c = 6

u_{1, 2} = \frac{- ( - 17 ) \pm ( - 17 ) ^{2} - 4 \cdot 12 \cdot 6}{2 \cdot 12}

u_{1, 2} = \frac{- ( - 17 ) \pm ( - 17 ) ^{2} - 4 \cdot 12 \cdot 6}{2 \cdot 12}

(- 17)^{2} - 4 \cdot 12 \cdot 6 = 1

(- 17)^{2} - 4 \cdot 12 \cdot 6

使用指数法则:

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

若

n

是偶数

(- 17)^{2} = 1 7^{2}

= 1 7^{2} - 4 \cdot 12 \cdot 6

数字相乘:

4 \cdot 12 \cdot 6 = 288

= 1 7^{2} - 288

1 7^{2} = 289

= 289 - 288

数字相减:

289 - 288 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- ( - 17 ) \pm 1}{2 \cdot 12}

将解分隔开

u_{1} = \frac{- ( - 17 ) + 1}{2 \cdot 12}, u_{2} = \frac{- ( - 17 ) - 1}{2 \cdot 12}

u = \frac{- ( - 17 ) + 1}{2 \cdot 12} : \frac{3}{4}

\frac{- ( - 17 ) + 1}{2 \cdot 12}

使用法则

- (- a) = a

= \frac{17 + 1}{2 \cdot 12}

数字相加:

17 + 1 = 18

= \frac{18}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{18}{24}

约分:

6

= \frac{3}{4}

u = \frac{- ( - 17 ) - 1}{2 \cdot 12} : \frac{2}{3}

\frac{- ( - 17 ) - 1}{2 \cdot 12}

使用法则

- (- a) = a

= \frac{17 - 1}{2 \cdot 12}

数字相减:

17 - 1 = 16

= \frac{16}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{16}{24}

约分:

8

= \frac{2}{3}

二次方程组的解是:

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

u = sin (x)

代回

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

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

sin (x) = \frac{3}{4} : x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

sin (x) = \frac{3}{4}

使用反三角函数性质

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

的通解

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

x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

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{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

以小数形式表示解

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

作图

流行的例子

sin(θ)=(sqrt(3))/3

sin (θ) = \frac{3}{3}

tan(x+pi)+cos(x+pi/2)=0

tan (x + π) + cos (x + \frac{π}{2}) = 0

sin(θ)cos(θ)=(sqrt(3))/4

sin (θ) cos (θ) = \frac{3}{4}

tan(x+pi/6)=sqrt(3)

tan (x + \frac{π}{6}) = 3

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