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受欢迎的三角函数 >

8(1-sin^2(x))+2sin(x)-7=0

解答

8 (1 - sin^{2} (x)) + 2 sin (x) - 7 = 0

解答

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \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

8 (1 - sin^{2} (x)) + 2 sin (x) - 7 = 0

用替代法求解

8 (1 - sin^{2} (x)) + 2 sin (x) - 7 = 0

令

: sin (x) = u

8 (1 - u^{2}) + 2 u - 7 = 0

8 (1 - u^{2}) + 2 u - 7 = 0 : u = - \frac{1}{4}, u = \frac{1}{2}

8 (1 - u^{2}) + 2 u - 7 = 0

展开

8 (1 - u^{2}) + 2 u - 7 : - 8 u^{2} + 2 u + 1

8 (1 - u^{2}) + 2 u - 7

乘开

8 (1 - u^{2}) : 8 - 8 u^{2}

8 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 8, b = 1, c = u^{2}

= 8 \cdot 1 - 8 u^{2}

数字相乘:

8 \cdot 1 = 8

= 8 - 8 u^{2}

= 8 - 8 u^{2} + 2 u - 7

化简

8 - 8 u^{2} + 2 u - 7 : - 8 u^{2} + 2 u + 1

8 - 8 u^{2} + 2 u - 7

对同类项分组

= - 8 u^{2} + 2 u + 8 - 7

数字相加/相减:

8 - 7 = 1

= - 8 u^{2} + 2 u + 1

= - 8 u^{2} + 2 u + 1

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 8, b = 2, c = 1

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

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

2^{2} - 4 (- 8) \cdot 1 = 6

2^{2} - 4 (- 8) \cdot 1

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 8 \cdot 1 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数字相加:

4 + 32 = 36

= 36

因式分解数字:

36 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a

6^{2} = 6

= 6

u_{1, 2} = \frac{- 2 \pm 6}{2 ( - 8 )}

将解分隔开

u_{1} = \frac{- 2 + 6}{2 ( - 8 )}, u_{2} = \frac{- 2 - 6}{2 ( - 8 )}

u = \frac{- 2 + 6}{2 ( - 8 )} : - \frac{1}{4}

\frac{- 2 + 6}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 2 + 6}{- 2 \cdot 8}

数字相加/相减:

- 2 + 6 = 4

= \frac{4}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{4}{- 16}

使用分式法则:

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

= - \frac{4}{16}

约分:

4

= - \frac{1}{4}

u = \frac{- 2 - 6}{2 ( - 8 )} : \frac{1}{2}

\frac{- 2 - 6}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 2 - 6}{- 2 \cdot 8}

数字相减:

- 2 - 6 = - 8

= \frac{- 8}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 8}{- 16}

使用分式法则:

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

= \frac{8}{16}

约分:

8

= \frac{1}{2}

二次方程组的解是:

u = - \frac{1}{4}, u = \frac{1}{2}

u = sin (x)

代回

sin (x) = - \frac{1}{4}, sin (x) = \frac{1}{2}

sin (x) = - \frac{1}{4}, sin (x) = \frac{1}{2}

sin (x) = - \frac{1}{4} : x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

sin (x) = - \frac{1}{4}

使用反三角函数性质

sin (x) = - \frac{1}{4}

sin (x) = - \frac{1}{4}

的通解

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

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 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}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

以小数形式表示解

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

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