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受欢迎的三角函数 >

tan(x+20)=cot(x)

解答

tan (x + 2 0^{\circ}) = cot (x)

解答

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}, x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

+1

弧度

x = \frac{π}{4} - \frac{π}{18} + πn, x = - \frac{π}{18} + \frac{3 π}{4} + πn

求解步骤

tan (x + 2 0^{\circ}) = cot (x)

两边减去

cot (x)

tan (x + 2 0^{\circ}) - cot (x) = 0

用 sin, cos 表示

- cot (x) + tan (2 0^{\circ} + x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

- \frac{cos ( x )}{sin ( x )} + \frac{sin ( 2 0 ^{\circ} + x )}{cos ( 2 0 ^{\circ} + x )} : \frac{- cos ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} ) + sin ( \frac{18 0 ^{\circ} + 9 x}{9} ) sin ( x )}{sin ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

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

\frac{sin ( 2 0 ^{\circ} + x )}{cos ( 2 0 ^{\circ} + x )} = \frac{sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

\frac{sin ( 2 0 ^{\circ} + x )}{cos ( 2 0 ^{\circ} + x )}

化简

2 0^{\circ} + x : \frac{18 0 ^{\circ} + 9 x}{9}

2 0^{\circ} + x

将项转换为分式:

x = \frac{x 9}{9}

= 2 0^{\circ} + \frac{x \cdot 9}{9}

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{18 0 ^{\circ} + x \cdot 9}{9}

= \frac{sin ( 2 0 ^{\circ} + x )}{cos ( \frac{18 0 ^{\circ} + x \cdot 9}{9} )}

化简

2 0^{\circ} + x : \frac{18 0 ^{\circ} + 9 x}{9}

2 0^{\circ} + x

将项转换为分式:

x = \frac{x 9}{9}

= 2 0^{\circ} + \frac{x \cdot 9}{9}

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{18 0 ^{\circ} + x \cdot 9}{9}

= \frac{sin ( \frac{18 0 ^{\circ} + x \cdot 9}{9} )}{cos ( \frac{18 0 ^{\circ} + x \cdot 9}{9} )}

= - \frac{cos ( x )}{sin ( x )} + \frac{sin ( \frac{9 x + 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

sin (x), cos (\frac{18 0 ^{\circ} + x 9}{9})

的最小公倍数:

sin (x) cos (\frac{9 x + 18 0 ^{\circ}}{9})

sin (x), cos (\frac{18 0 ^{\circ} + x \cdot 9}{9})

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (\frac{18 0 ^{\circ} + x 9}{9})

中的因子组成的表达式

= sin (x) cos (\frac{9 x + 18 0 ^{\circ}}{9})

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{9 x + 18 0 ^{\circ}}{9})

对于

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

将分母和分子乘以

cos (\frac{9 x + 18 0 ^{\circ}}{9})

\frac{cos ( x )}{sin ( x )} = \frac{cos ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}{sin ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

对于

\frac{sin ( \frac{18 0 ^{\circ} + x \cdot 9}{9} )}{cos ( \frac{18 0 ^{\circ} + x \cdot 9}{9} )} :

将分母和分子乘以

sin (x)

\frac{sin ( \frac{18 0 ^{\circ} + x \cdot 9}{9} )}{cos ( \frac{18 0 ^{\circ} + x \cdot 9}{9} )} = \frac{sin ( \frac{18 0 ^{\circ} + x \cdot 9}{9} ) sin ( x )}{cos ( \frac{18 0 ^{\circ} + x \cdot 9}{9} ) sin ( x )}

= - \frac{cos ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}{sin ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )} + \frac{sin ( \frac{18 0 ^{\circ} + x \cdot 9}{9} ) sin ( x )}{cos ( \frac{18 0 ^{\circ} + x \cdot 9}{9} ) 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 ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} ) + sin ( \frac{18 0 ^{\circ} + x \cdot 9}{9} ) sin ( x )}{sin ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

= \frac{- cos ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} ) + sin ( \frac{18 0 ^{\circ} + 9 x}{9} ) sin ( x )}{sin ( x ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

\frac{- cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) cos ( x ) + sin ( \frac{18 0 ^{\circ} + 9 x}{9} ) sin ( x )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) sin ( x )} = 0

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

- cos (\frac{18 0 ^{\circ} + 9 x}{9}) cos (x) + sin (\frac{18 0 ^{\circ} + 9 x}{9}) sin (x) = 0

使用三角恒等式改写

- cos (\frac{18 0 ^{\circ} + 9 x}{9}) cos (x) + sin (\frac{18 0 ^{\circ} + 9 x}{9}) sin (x)

使用角和恒等式:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

- cos (s) cos (t) + sin (s) sin (t) = - cos (s + t)

= - cos (\frac{18 0 ^{\circ} + 9 x}{9} + x)

- cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

两边除以

- 1

- cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

两边除以

- 1

\frac{- cos ( \frac{18 0 ^{\circ} + 9 x}{9} + x )}{- 1} = \frac{0}{- 1}

化简

cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

cos (\frac{18 0 ^{\circ} + 9 x}{9} + x) = 0

的通解

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n, \frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n, \frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

解

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n

在两边乘以

9

\frac{18 0 ^{\circ} + 9 x}{9} + x = 9 0^{\circ} + 36 0^{\circ} n

在两边乘以

9

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 + x \cdot 9 = 9 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

化简

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 + x \cdot 9 = 9 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

化简

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 : 18 0^{\circ} + 9 x

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9

分式相乘:

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

= \frac{( 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

约分:

9

= 18 0^{\circ} + 9 x

化简

x \cdot 9 : 9 x

x \cdot 9

使用交换律:

x \cdot 9 = 9 x

9 x

化简

9 0^{\circ} \cdot 9 : 81 0^{\circ}

9 0^{\circ} \cdot 9

分式相乘:

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

= 81 0^{\circ}

化简

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

数字相乘:

2 \cdot 9 = 18

= 324 0^{\circ} n

18 0^{\circ} + 9 x + 9 x = 81 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

将

18 0^{\circ}

到右边

18 0^{\circ} + 18 x = 81 0^{\circ} + 324 0^{\circ} n

两边减去

18 0^{\circ}

18 0^{\circ} + 18 x - 18 0^{\circ} = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

化简

18 x = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

18 x = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

两边除以

18

18 x = 81 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

两边除以

18

\frac{18 x}{18} = \frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

化简

\frac{18 x}{18} = \frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

化简

\frac{18 x}{18} : x

\frac{18 x}{18}

数字相除:

\frac{18}{18} = 1

= x

化简

\frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ} : 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

\frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

对同类项分组

= - 1 0^{\circ} + \frac{324 0 ^{\circ} n}{18} + \frac{81 0 ^{\circ}}{18}

\frac{324 0 ^{\circ} n}{18} = 18 0^{\circ} n

\frac{324 0 ^{\circ} n}{18}

数字相除:

\frac{18}{18} = 1

= 18 0^{\circ} n

\frac{81 0 ^{\circ}}{18} = 4 5^{\circ}

\frac{81 0 ^{\circ}}{18}

使用分式法则:

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

= \frac{162 0 ^{\circ}}{2 \cdot 18}

数字相乘:

2 \cdot 18 = 36

= 4 5^{\circ}

约分:

9

= 4 5^{\circ}

= - 1 0^{\circ} + 18 0^{\circ} n + 4 5^{\circ}

对同类项分组

= 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}

解

\frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

\frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

在两边乘以

9

\frac{18 0 ^{\circ} + 9 x}{9} + x = 27 0^{\circ} + 36 0^{\circ} n

在两边乘以

9

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 + x \cdot 9 = 27 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

化简

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 + x \cdot 9 = 27 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

化简

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 : 18 0^{\circ} + 9 x

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9

分式相乘:

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

= \frac{( 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

约分:

9

= 18 0^{\circ} + 9 x

化简

x \cdot 9 : 9 x

x \cdot 9

使用交换律:

x \cdot 9 = 9 x

9 x

化简

27 0^{\circ} \cdot 9 : 243 0^{\circ}

27 0^{\circ} \cdot 9

分式相乘:

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

= 243 0^{\circ}

数字相乘:

3 \cdot 9 = 27

= 243 0^{\circ}

化简

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

数字相乘:

2 \cdot 9 = 18

= 324 0^{\circ} n

18 0^{\circ} + 9 x + 9 x = 243 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

将

18 0^{\circ}

到右边

18 0^{\circ} + 18 x = 243 0^{\circ} + 324 0^{\circ} n

两边减去

18 0^{\circ}

18 0^{\circ} + 18 x - 18 0^{\circ} = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

化简

18 x = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

18 x = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

两边除以

18

18 x = 243 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

两边除以

18

\frac{18 x}{18} = \frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

化简

\frac{18 x}{18} = \frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

化简

\frac{18 x}{18} : x

\frac{18 x}{18}

数字相除:

\frac{18}{18} = 1

= x

化简

\frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ} : 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

\frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} - 1 0^{\circ}

对同类项分组

= - 1 0^{\circ} + \frac{324 0 ^{\circ} n}{18} + \frac{243 0 ^{\circ}}{18}

\frac{324 0 ^{\circ} n}{18} = 18 0^{\circ} n

\frac{324 0 ^{\circ} n}{18}

数字相除:

\frac{18}{18} = 1

= 18 0^{\circ} n

\frac{243 0 ^{\circ}}{18} = 13 5^{\circ}

\frac{243 0 ^{\circ}}{18}

使用分式法则:

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

= \frac{486 0 ^{\circ}}{2 \cdot 18}

数字相乘:

2 \cdot 18 = 36

= 13 5^{\circ}

约分:

9

= 13 5^{\circ}

= - 1 0^{\circ} + 18 0^{\circ} n + 13 5^{\circ}

对同类项分组

= 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

x = 18 0^{\circ} n + 4 5^{\circ} - 1 0^{\circ}, x = 18 0^{\circ} n - 1 0^{\circ} + 13 5^{\circ}

作图

流行的例子

3 csc^{2} (x) + 1 = 5

6cos^2(x)-12cos(x)+6=0

6 cos^{2} (x) - 12 cos (x) + 6 = 0

6sin^2(θ)-10sin(θ)-8=-3sin(θ)-3

6 sin^{2} (θ) - 10 sin (θ) - 8 = - 3 sin (θ) - 3

sin(4θ)-sin(6θ)=0

sin (4 θ) - sin (6 θ) = 0

sin (x) = \frac{3}{10}