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受欢迎的三角函数 >

2sin^2(x)-15sin(x)+7=0

解答

2 sin^{2} (x) - 15 sin (x) + 7 = 0

解答

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

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

2 sin^{2} (x) - 15 sin (x) + 7 = 0

用替代法求解

2 sin^{2} (x) - 15 sin (x) + 7 = 0

令

: sin (x) = u

2 u^{2} - 15 u + 7 = 0

2 u^{2} - 15 u + 7 = 0 : u = 7, u = \frac{1}{2}

2 u^{2} - 15 u + 7 = 0

使用求根公式求解

2 u^{2} - 15 u + 7 = 0

二次方程求根公式:

若

a = 2, b = - 15, c = 7

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 \cdot 2 \cdot 7}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 \cdot 2 \cdot 7}{2 \cdot 2}

(- 15)^{2} - 4 \cdot 2 \cdot 7 = 13

(- 15)^{2} - 4 \cdot 2 \cdot 7

使用指数法则:

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

若

n

是偶数

(- 15)^{2} = 1 5^{2}

= 1 5^{2} - 4 \cdot 2 \cdot 7

数字相乘:

4 \cdot 2 \cdot 7 = 56

= 1 5^{2} - 56

1 5^{2} = 225

= 225 - 56

数字相减:

225 - 56 = 169

= 169

因式分解数字:

169 = 1 3^{2}

= 1 3^{2}

使用根式运算法则:

n a^{n} = a

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- ( - 15 ) \pm 13}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 15 ) + 13}{2 \cdot 2}, u_{2} = \frac{- ( - 15 ) - 13}{2 \cdot 2}

u = \frac{- ( - 15 ) + 13}{2 \cdot 2} : 7

\frac{- ( - 15 ) + 13}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{15 + 13}{2 \cdot 2}

数字相加:

15 + 13 = 28

= \frac{28}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{28}{4}

数字相除:

\frac{28}{4} = 7

= 7

u = \frac{- ( - 15 ) - 13}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 15 ) - 13}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{15 - 13}{2 \cdot 2}

数字相减:

15 - 13 = 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

约分:

2

= \frac{1}{2}

二次方程组的解是:

u = 7, u = \frac{1}{2}

u = sin (x)

代回

sin (x) = 7, sin (x) = \frac{1}{2}

sin (x) = 7, sin (x) = \frac{1}{2}

sin (x) = 7 :

无解

sin (x) = 7

- 1 \leq sin (x) \leq 1

无解

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

作图

流行的例子

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