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受欢迎的三角函数 >

2tan(x/2)-csc(x)=0

解答

2 tan (\frac{x}{2}) - csc (x) = 0

解答

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

+1

度数

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n

求解步骤

2 tan (\frac{x}{2}) - csc (x) = 0

用 sin, cos 表示

- csc (x) + 2 tan (\frac{x}{2})

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

乘

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

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

分式相乘:

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

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

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

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

的最小公倍数:

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

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

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (\frac{x}{2})

中的因子组成的表达式

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

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

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

对于

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

将分母和分子乘以

cos (\frac{x}{2})

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

对于

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

将分母和分子乘以

sin (x)

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

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

因为分母相等，所以合并分式:

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

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

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

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

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

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

使用三角恒等式改写

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

使用积化和差恒等式:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

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

化简

- cos (\frac{x}{2}) + 2 \cdot \frac{1}{2} (cos (\frac{x}{2} - x) - cos (\frac{x}{2} + x)) : - cos (\frac{3 x}{2})

- cos (\frac{x}{2}) + 2 \cdot \frac{1}{2} (cos (\frac{x}{2} - x) - cos (\frac{x}{2} + x))

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

2 \cdot \frac{1}{2} (cos (\frac{x}{2} - x) - cos (\frac{x}{2} + x))

分式相乘:

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

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

约分:

2

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

化简

\frac{x}{2} - x : - \frac{x}{2}

\frac{x}{2} - x

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x}{2} - \frac{x \cdot 2}{2}

因为分母相等，所以合并分式:

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

= \frac{x - x \cdot 2}{2}

同类项相加:

x - 2 x = - x

= \frac{- x}{2}

使用分式法则:

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

= - \frac{x}{2}

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

化简

\frac{x}{2} + x : \frac{3 x}{2}

\frac{x}{2} + x

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x}{2} + \frac{x \cdot 2}{2}

因为分母相等，所以合并分式:

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

= \frac{x + x \cdot 2}{2}

同类项相加:

x + 2 x = 3 x

= \frac{3 x}{2}

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

整理后得

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

使用负角恒等式:

cos (- x) = cos (x)

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

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

同类项相加:

- cos (\frac{x}{2}) + cos (\frac{x}{2}) = 0

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

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

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

两边除以

- 1

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

两边除以

- 1

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

化简

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

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

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

的通解

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

\frac{3 x}{2} = \frac{π}{2} + 2 πn, \frac{3 x}{2} = \frac{3 π}{2} + 2 πn

\frac{3 x}{2} = \frac{π}{2} + 2 πn, \frac{3 x}{2} = \frac{3 π}{2} + 2 πn

解

\frac{3 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{3} + \frac{4 πn}{3}

\frac{3 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

数字相乘:

2 \cdot 3 = 6

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 x

化简

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分式相乘:

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

= \frac{π 2}{2}

约分:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

两边除以

3

3 x = π + 4 πn

两边除以

3

\frac{3 x}{3} = \frac{π}{3} + \frac{4 πn}{3}

化简

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

解

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn : x = π + \frac{4 πn}{3}

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

数字相乘:

2 \cdot 3 = 6

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 x

化简

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

分式相乘:

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

= \frac{3 π 2}{2}

约分:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

两边除以

3

3 x = 3 π + 4 πn

两边除以

3

\frac{3 x}{3} = \frac{3 π}{3} + \frac{4 πn}{3}

化简

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

作图

流行的例子

sin^2(x)-sin(x)=2

sin^{2} (x) - sin (x) = 2

cos(4x)=(sqrt(2))/2

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

2sin(θ)=-sqrt(2)

2 sin (θ) = - 2

sqrt(3)cot(t)-1=0

3 cot (t) - 1 = 0

2cos^2(x)+3sin(x)=3

2 cos^{2} (x) + 3 sin (x) = 3