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受欢迎的三角函数 >

(sin(2x))/(4sin(2x)-1)=1

解答

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

解答

x = \frac{0.33983 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.33983 \dots}{2} + πn

+1

度数

x = 9.73561 \dots^{\circ} + 18 0^{\circ} n, x = 80.26438 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (2 x) = u

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

\frac{u}{4 u - 1} = 1 : u = \frac{1}{3}

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

在两边乘以

4 u - 1

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

在两边乘以

4 u - 1

\frac{u}{4 u - 1} (4 u - 1) = 1 \cdot (4 u - 1)

化简

\frac{u}{4 u - 1} (4 u - 1) = 1 \cdot (4 u - 1)

化简

\frac{u}{4 u - 1} (4 u - 1) : u

\frac{u}{4 u - 1} (4 u - 1)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{u ( 4 u - 1 )}{4 u - 1}

约分:

4 u - 1

= u

化简

1 \cdot (4 u - 1) : 4 u - 1

1 \cdot (4 u - 1)

乘以:

1 \cdot (4 u - 1) = (4 u - 1)

= (4 u - 1)

去除括号:

(a) = a

= 4 u - 1

u = 4 u - 1

u = 4 u - 1

u = 4 u - 1

将

4 u

para o lado esquerdo

u = 4 u - 1

两边减去

4 u

u - 4 u = 4 u - 1 - 4 u

化简

- 3 u = - 1

- 3 u = - 1

两边除以

- 3

- 3 u = - 1

两边除以

- 3

\frac{- 3 u}{- 3} = \frac{- 1}{- 3}

化简

u = \frac{1}{3}

u = \frac{1}{3}

验证解

找到无定义的点（奇点）:

u = \frac{1}{4}

取

\frac{u}{4 u - 1}

的分母，令其等于零

解

4 u - 1 = 0 : u = \frac{1}{4}

4 u - 1 = 0

将

1

到右边

4 u - 1 = 0

两边加上

1

4 u - 1 + 1 = 0 + 1

化简

4 u = 1

4 u = 1

两边除以

4

4 u = 1

两边除以

4

\frac{4 u}{4} = \frac{1}{4}

化简

u = \frac{1}{4}

u = \frac{1}{4}

以下点无定义

u = \frac{1}{4}

将不在定义域的点与解相综合:

u = \frac{1}{3}

u = sin (2 x)

代回

sin (2 x) = \frac{1}{3}

sin (2 x) = \frac{1}{3}

sin (2 x) = \frac{1}{3} : x = \frac{arcsin ( \frac{1}{3} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + πn

sin (2 x) = \frac{1}{3}

使用反三角函数性质

sin (2 x) = \frac{1}{3}

sin (2 x) = \frac{1}{3}

的通解

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

2 x = arcsin (\frac{1}{3}) + 2 πn, 2 x = π - arcsin (\frac{1}{3}) + 2 πn

2 x = arcsin (\frac{1}{3}) + 2 πn, 2 x = π - arcsin (\frac{1}{3}) + 2 πn

解

2 x = arcsin (\frac{1}{3}) + 2 πn : x = \frac{arcsin ( \frac{1}{3} )}{2} + πn

2 x = arcsin (\frac{1}{3}) + 2 πn

两边除以

2

2 x = arcsin (\frac{1}{3}) + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

化简

x = \frac{arcsin ( \frac{1}{3} )}{2} + πn

x = \frac{arcsin ( \frac{1}{3} )}{2} + πn

解

2 x = π - arcsin (\frac{1}{3}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + πn

2 x = π - arcsin (\frac{1}{3}) + 2 πn

两边除以

2

2 x = π - arcsin (\frac{1}{3}) + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

化简

x = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + πn

x = \frac{arcsin ( \frac{1}{3} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + πn

合并所有解

x = \frac{arcsin ( \frac{1}{3} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + πn

以小数形式表示解

x = \frac{0.33983 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.33983 \dots}{2} + πn

作图

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