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受欢迎的三角函数 >

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

解答

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

解答

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 199.47122 \dots^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

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

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

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

改写成标准形式

a x^{2} + b x + c = 0

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

1^{2} - 4 (- 6) \cdot 1 = 5

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 6) \cdot 1

使用法则

- (- a) = a

= 1 + 4 \cdot 6 \cdot 1

数字相乘:

4 \cdot 6 \cdot 1 = 24

= 1 + 24

数字相加:

1 + 24 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 ( - 6 )}

将解分隔开

u_{1} = \frac{- 1 + 5}{2 ( - 6 )}, u_{2} = \frac{- 1 - 5}{2 ( - 6 )}

u = \frac{- 1 + 5}{2 ( - 6 )} : - \frac{1}{3}

\frac{- 1 + 5}{2 ( - 6 )}

去除括号:

(- a) = - a

= \frac{- 1 + 5}{- 2 \cdot 6}

数字相加/相减:

- 1 + 5 = 4

= \frac{4}{- 2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{4}{- 12}

使用分式法则:

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

= - \frac{4}{12}

约分:

4

= - \frac{1}{3}

u = \frac{- 1 - 5}{2 ( - 6 )} : \frac{1}{2}

\frac{- 1 - 5}{2 ( - 6 )}

去除括号:

(- a) = - a

= \frac{- 1 - 5}{- 2 \cdot 6}

数字相减:

- 1 - 5 = - 6

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

数字相乘:

2 \cdot 6 = 12

= \frac{- 6}{- 12}

使用分式法则:

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

= \frac{6}{12}

约分:

6

= \frac{1}{2}

二次方程组的解是:

u = - \frac{1}{3}, u = \frac{1}{2}

u = sin (x)

代回

sin (x) = - \frac{1}{3}, sin (x) = \frac{1}{2}

sin (x) = - \frac{1}{3}, sin (x) = \frac{1}{2}

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

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

合并所有解

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

以小数形式表示解

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

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