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受欢迎的三角函数 >

20sin^2(x)-26sin(x)+8=0

解答

20 sin^{2} (x) - 26 sin (x) + 8 = 0

解答

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \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

20 sin^{2} (x) - 26 sin (x) + 8 = 0

用替代法求解

20 sin^{2} (x) - 26 sin (x) + 8 = 0

令

: sin (x) = u

20 u^{2} - 26 u + 8 = 0

20 u^{2} - 26 u + 8 = 0 : u = \frac{4}{5}, u = \frac{1}{2}

20 u^{2} - 26 u + 8 = 0

使用求根公式求解

20 u^{2} - 26 u + 8 = 0

二次方程求根公式:

若

a = 20, b = - 26, c = 8

u_{1, 2} = \frac{- ( - 26 ) \pm ( - 26 ) ^{2} - 4 \cdot 20 \cdot 8}{2 \cdot 20}

u_{1, 2} = \frac{- ( - 26 ) \pm ( - 26 ) ^{2} - 4 \cdot 20 \cdot 8}{2 \cdot 20}

(- 26)^{2} - 4 \cdot 20 \cdot 8 = 6

(- 26)^{2} - 4 \cdot 20 \cdot 8

使用指数法则:

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

若

n

是偶数

(- 26)^{2} = 2 6^{2}

= 2 6^{2} - 4 \cdot 20 \cdot 8

数字相乘:

4 \cdot 20 \cdot 8 = 640

= 2 6^{2} - 640

2 6^{2} = 676

= 676 - 640

数字相减:

676 - 640 = 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{- ( - 26 ) \pm 6}{2 \cdot 20}

将解分隔开

u_{1} = \frac{- ( - 26 ) + 6}{2 \cdot 20}, u_{2} = \frac{- ( - 26 ) - 6}{2 \cdot 20}

u = \frac{- ( - 26 ) + 6}{2 \cdot 20} : \frac{4}{5}

\frac{- ( - 26 ) + 6}{2 \cdot 20}

使用法则

- (- a) = a

= \frac{26 + 6}{2 \cdot 20}

数字相加:

26 + 6 = 32

= \frac{32}{2 \cdot 20}

数字相乘:

2 \cdot 20 = 40

= \frac{32}{40}

约分:

8

= \frac{4}{5}

u = \frac{- ( - 26 ) - 6}{2 \cdot 20} : \frac{1}{2}

\frac{- ( - 26 ) - 6}{2 \cdot 20}

使用法则

- (- a) = a

= \frac{26 - 6}{2 \cdot 20}

数字相减:

26 - 6 = 20

= \frac{20}{2 \cdot 20}

数字相乘:

2 \cdot 20 = 40

= \frac{20}{40}

约分:

20

= \frac{1}{2}

二次方程组的解是:

u = \frac{4}{5}, u = \frac{1}{2}

u = sin (x)

代回

sin (x) = \frac{4}{5}, sin (x) = \frac{1}{2}

sin (x) = \frac{4}{5}, sin (x) = \frac{1}{2}

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

sin (x) = \frac{4}{5}

使用反三角函数性质

sin (x) = \frac{4}{5}

sin (x) = \frac{4}{5}

的通解

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

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

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

以小数形式表示解

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

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