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受欢迎的三角函数 >

((2sin(x)-1))/((sin(x)+5))=0

解答

\frac{( 2 sin ( x ) - 1 )}{( sin ( x ) + 5 )} = 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

\frac{( 2 sin ( x ) - 1 )}{( sin ( x ) + 5 )} = 0

用替代法求解

\frac{2 sin ( x ) - 1}{sin ( x ) + 5} = 0

令

: sin (x) = u

\frac{2 u - 1}{u + 5} = 0

\frac{2 u - 1}{u + 5} = 0 : u = \frac{1}{2}

\frac{2 u - 1}{u + 5} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 u - 1 = 0

将

1

到右边

2 u - 1 = 0

两边加上

1

2 u - 1 + 1 = 0 + 1

化简

2 u = 1

2 u = 1

两边除以

2

2 u = 1

两边除以

2

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

化简

u = \frac{1}{2}

u = \frac{1}{2}

验证解

找到无定义的点（奇点）:

u = - 5

取

\frac{2 u - 1}{u + 5}

的分母，令其等于零

解

u + 5 = 0 : u = - 5

u + 5 = 0

将

5

到右边

u + 5 = 0

两边减去

5

u + 5 - 5 = 0 - 5

化简

u = - 5

u = - 5

以下点无定义

u = - 5

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

流行的例子

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