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受欢迎的三角函数 >

(5sin(x)+6)/(sin(x))=17

解答

\frac{5 sin ( x ) + 6}{sin ( x )} = 17

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

\frac{5 sin ( x ) + 6}{sin ( x )} = 17

用替代法求解

\frac{5 sin ( x ) + 6}{sin ( x )} = 17

令

: sin (x) = u

\frac{5 u + 6}{u} = 17

\frac{5 u + 6}{u} = 17 : u = \frac{1}{2}

\frac{5 u + 6}{u} = 17

在两边乘以

u

\frac{5 u + 6}{u} = 17

在两边乘以

u

\frac{5 u + 6}{u} u = 17 u

化简

5 u + 6 = 17 u

5 u + 6 = 17 u

将

6

到右边

5 u + 6 = 17 u

两边减去

6

5 u + 6 - 6 = 17 u - 6

化简

5 u = 17 u - 6

5 u = 17 u - 6

将

17 u

para o lado esquerdo

5 u = 17 u - 6

两边减去

17 u

5 u - 17 u = 17 u - 6 - 17 u

化简

- 12 u = - 6

- 12 u = - 6

两边除以

- 12

- 12 u = - 6

两边除以

- 12

\frac{- 12 u}{- 12} = \frac{- 6}{- 12}

化简

\frac{- 12 u}{- 12} = \frac{- 6}{- 12}

化简

\frac{- 12 u}{- 12} : u

\frac{- 12 u}{- 12}

使用分式法则:

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

= \frac{12 u}{12}

数字相除:

\frac{12}{12} = 1

= u

化简

\frac{- 6}{- 12} : \frac{1}{2}

\frac{- 6}{- 12}

使用分式法则:

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

= \frac{6}{12}

约分:

6

= \frac{1}{2}

u = \frac{1}{2}

u = \frac{1}{2}

u = \frac{1}{2}

验证解

找到无定义的点（奇点）:

u = 0

取

\frac{5 u + 6}{u}

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{1}{2}

u = sin (x)

代回

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

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

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

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