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受欢迎的三角函数 >

2sin^2(x)-7sin(x)-4=0

解答

2 sin^{2} (x) - 7 sin (x) - 4 = 0

解答

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

2 sin^{2} (x) - 7 sin (x) - 4 = 0

用替代法求解

2 sin^{2} (x) - 7 sin (x) - 4 = 0

令

: sin (x) = u

2 u^{2} - 7 u - 4 = 0

2 u^{2} - 7 u - 4 = 0 : u = 4, u = - \frac{1}{2}

2 u^{2} - 7 u - 4 = 0

使用求根公式求解

2 u^{2} - 7 u - 4 = 0

二次方程求根公式:

若

a = 2, b = - 7, c = - 4

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 (- 4) = 9

(- 7)^{2} - 4 \cdot 2 (- 4)

使用法则

- (- a) = a

= (- 7)^{2} + 4 \cdot 2 \cdot 4

使用指数法则:

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

若

n

是偶数

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

= 7^{2} + 4 \cdot 2 \cdot 4

数字相乘:

4 \cdot 2 \cdot 4 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

数字相加:

49 + 32 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 7 ) \pm 9}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 9}{2 \cdot 2}, u_{2} = \frac{- ( - 7 ) - 9}{2 \cdot 2}

u = \frac{- ( - 7 ) + 9}{2 \cdot 2} : 4

\frac{- ( - 7 ) + 9}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{7 + 9}{2 \cdot 2}

数字相加:

7 + 9 = 16

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

数字相乘:

2 \cdot 2 = 4

= \frac{16}{4}

数字相除:

\frac{16}{4} = 4

= 4

u = \frac{- ( - 7 ) - 9}{2 \cdot 2} : - \frac{1}{2}

\frac{- ( - 7 ) - 9}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{7 - 9}{2 \cdot 2}

数字相减:

7 - 9 = - 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

二次方程组的解是:

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

u = sin (x)

代回

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

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

sin (x) = 4 :

无解

sin (x) = 4

- 1 \leq sin (x) \leq 1

无解

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

合并所有解

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

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