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受欢迎的三角函数 >

(cos(2x))/(sin(3x)-sin(x))-1=0

解答

\frac{cos ( 2 x )}{sin ( 3 x ) - sin ( x )} - 1 = 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{cos ( 2 x )}{sin ( 3 x ) - sin ( x )} - 1 = 0

使用三角恒等式改写

\frac{cos ( 2 x )}{sin ( 3 x ) - sin ( x )} - 1

使用和差化积恒等式:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

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

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

\frac{cos ( 2 x )}{2 sin ( \frac{3 x - x}{2} ) cos ( \frac{3 x + x}{2} )}

2 sin (\frac{3 x - x}{2}) cos (\frac{3 x + x}{2}) = 2 sin (\frac{2 x}{2}) cos (\frac{4 x}{2})

2 sin (\frac{3 x - x}{2}) cos (\frac{3 x + x}{2})

同类项相加:

3 x - x = 2 x

= 2 sin (\frac{2 x}{2}) cos (\frac{3 x + x}{2})

同类项相加:

3 x + x = 4 x

= 2 sin (\frac{2 x}{2}) cos (\frac{4 x}{2})

= \frac{cos ( 2 x )}{2 sin ( \frac{2 x}{2} ) cos ( \frac{4 x}{2} )}

数字相除:

\frac{2}{2} = 1

= \frac{cos ( 2 x )}{2 sin ( x ) cos ( \frac{4 x}{2} )}

数字相除:

\frac{4}{2} = 2

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

约分:

cos (2 x)

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

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

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

在两边乘以

2 sin (x)

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

在两边乘以

2 sin (x)

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

化简

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

化简

\frac{1}{2 sin ( x )} \cdot 2 sin (x) : 1

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

分式相乘:

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

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

约分:

2

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

约分:

sin (x)

= 1

化简

- 1 \cdot 2 sin (x) : - 2 sin (x)

- 1 \cdot 2 sin (x)

数字相乘:

1 \cdot 2 = 2

= - 2 sin (x)

化简

0 \cdot 2 sin (x) : 0

0 \cdot 2 sin (x)

使用法则

0 \cdot a = 0

= 0

1 - 2 sin (x) = 0

1 - 2 sin (x) = 0

1 - 2 sin (x) = 0

将

1

到右边

1 - 2 sin (x) = 0

两边减去

1

1 - 2 sin (x) - 1 = 0 - 1

化简

- 2 sin (x) = - 1

- 2 sin (x) = - 1

两边除以

- 2

- 2 sin (x) = - 1

两边除以

- 2

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

化简

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

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

验证解

找到无定义的点（奇点）:

sin (x) = 0

取

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

的分母，令其等于零

解

2 sin (x) = 0 : sin (x) = 0

2 sin (x) = 0

两边除以

2

2 sin (x) = 0

两边除以

2

\frac{2 sin ( x )}{2} = \frac{0}{2}

化简

sin (x) = 0

sin (x) = 0

以下点无定义

sin (x) = 0

将不在定义域的点与解相综合:

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

作图

流行的例子

6cos(2x)-cos(x)-1=0

6 cos (2 x) - cos (x) - 1 = 0

8cos^2(x)-10sin(x)-11=0

8 cos^{2} (x) - 10 sin (x) - 11 = 0

6sin(x)=cos(x)-2

6 sin (x) = cos (x) - 2

csc (3 x) - 2 = 0

cos(20)+14sin^2(θ)=10

cos (2 0^{\circ}) + 14 sin^{2} (θ) = 10