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受欢迎的三角函数 >

tan(2x+13)=cot(x+53)

解答

tan (2 x + 13) = cot (x + 5 3^{\circ})

解答

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}, x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

+1

弧度

x = \frac{37 π}{540} - \frac{13}{3} + \frac{2 π}{3} n, x = \frac{217 π}{540} - \frac{13}{3} + \frac{2 π}{3} n

求解步骤

tan (2 x + 13) = cot (x + 5 3^{\circ})

两边减去

cot (x + 5 3^{\circ})

tan (2 x + 13) - cot (x + 5 3^{\circ}) = 0

用 sin, cos 表示

- cot (5 3^{\circ} + x) + tan (13 + 2 x)

使用基本三角恒等式:

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

= - \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + tan (13 + 2 x)

使用基本三角恒等式:

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

= - \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + \frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )}

化简

- \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + \frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )} : \frac{- cos ( \frac{954 0 ^{\circ} + 180 x}{180} ) cos ( 2 x + 13 ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} ) cos ( 2 x + 13 )}

- \frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} + \frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )}

\frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )} = \frac{cos ( \frac{954 0 ^{\circ} + 180 x}{180} )}{sin ( \frac{954 0 ^{\circ} + 180 x}{180} )}

\frac{cos ( 5 3 ^{\circ} + x )}{sin ( 5 3 ^{\circ} + x )}

化简

5 3^{\circ} + x : \frac{954 0 ^{\circ} + 180 x}{180}

5 3^{\circ} + x

将项转换为分式:

x = \frac{x 180}{180}

= 5 3^{\circ} + \frac{x \cdot 180}{180}

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{954 0 ^{\circ} + x \cdot 180}{180}

= \frac{cos ( 5 3 ^{\circ} + x )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}

化简

5 3^{\circ} + x : \frac{954 0 ^{\circ} + 180 x}{180}

5 3^{\circ} + x

将项转换为分式:

x = \frac{x 180}{180}

= 5 3^{\circ} + \frac{x \cdot 180}{180}

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{954 0 ^{\circ} + x \cdot 180}{180}

= \frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}

= - \frac{cos ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} )} + \frac{sin ( 2 x + 13 )}{cos ( 2 x + 13 )}

sin (\frac{954 0 ^{\circ} + x 180}{180}), cos (13 + 2 x)

的最小公倍数:

sin (\frac{180 x + 954 0 ^{\circ}}{180}) cos (2 x + 13)

sin (\frac{954 0 ^{\circ} + x \cdot 180}{180}), cos (13 + 2 x)

最小公倍数 (LCM)

计算出由出现在

sin (\frac{954 0 ^{\circ} + x 180}{180})

或

cos (13 + 2 x)

中的因子组成的表达式

= sin (\frac{180 x + 954 0 ^{\circ}}{180}) cos (2 x + 13)

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 (\frac{180 x + 954 0 ^{\circ}}{180}) cos (2 x + 13)

对于

\frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )} :

将分母和分子乘以

cos (2 x + 13)

\frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} )} = \frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )}

对于

\frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )} :

将分母和分子乘以

sin (\frac{180 x + 954 0 ^{\circ}}{180})

\frac{sin ( 13 + 2 x )}{cos ( 13 + 2 x )} = \frac{sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{cos ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}

= - \frac{cos ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )}{sin ( \frac{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 )} + \frac{sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{cos ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}

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{954 0 ^{\circ} + x \cdot 180}{180} ) cos ( 2 x + 13 ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} ) cos ( 2 x + 13 )}

= \frac{- cos ( \frac{954 0 ^{\circ} + 180 x}{180} ) cos ( 2 x + 13 ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{sin ( \frac{180 x + 954 0 ^{\circ}}{180} ) cos ( 2 x + 13 )}

\frac{- cos ( 13 + 2 x ) cos ( \frac{180 x + 954 0 ^{\circ}}{180} ) + sin ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )}{cos ( 13 + 2 x ) sin ( \frac{180 x + 954 0 ^{\circ}}{180} )} = 0

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

- cos (13 + 2 x) cos (\frac{180 x + 954 0 ^{\circ}}{180}) + sin (13 + 2 x) sin (\frac{180 x + 954 0 ^{\circ}}{180}) = 0

使用三角恒等式改写

- cos (13 + 2 x) cos (\frac{180 x + 954 0 ^{\circ}}{180}) + sin (13 + 2 x) sin (\frac{180 x + 954 0 ^{\circ}}{180})

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 (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180})

- cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

两边除以

- 1

- cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

两边除以

- 1

\frac{- cos ( 13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} )}{- 1} = \frac{0}{- 1}

化简

cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 0

cos (13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180}) = 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}

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n, 13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n, 13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

解

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n : x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n

将

13

到右边

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n

两边减去

13

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} - 13 = 9 0^{\circ} + 36 0^{\circ} n - 13

化简

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n - 13

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n - 13

在两边乘以

180

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 9 0^{\circ} + 36 0^{\circ} n - 13

在两边乘以

180

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 9 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

化简

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 9 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

化简

2 x \cdot 180 : 360 x

2 x \cdot 180

数字相乘:

2 \cdot 180 = 360

= 360 x

化简

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 : 180 x + 954 0^{\circ}

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180

分式相乘:

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

= \frac{( 180 x + 954 0 ^{\circ} ) \cdot 180}{180}

约分:

180

= 180 x + 954 0^{\circ}

化简

9 0^{\circ} \cdot 180 : 1620 0^{\circ}

9 0^{\circ} \cdot 180

分式相乘:

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

= 1620 0^{\circ}

数字相除:

\frac{180}{2} = 90

= 1620 0^{\circ}

化简

36 0^{\circ} n \cdot 180 : 6480 0^{\circ} n

36 0^{\circ} n \cdot 180

数字相乘:

2 \cdot 180 = 360

= 6480 0^{\circ} n

化简

- 13 \cdot 180 : - 2340

- 13 \cdot 180

数字相乘:

13 \cdot 180 = 2340

= - 2340

360 x + 180 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

将

954 0^{\circ}

到右边

540 x + 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340

两边减去

954 0^{\circ}

540 x + 954 0^{\circ} - 954 0^{\circ} = 1620 0^{\circ} + 6480 0^{\circ} n - 2340 - 954 0^{\circ}

化简

540 x = 666 0^{\circ} + 6480 0^{\circ} n - 2340

540 x = 666 0^{\circ} + 6480 0^{\circ} n - 2340

两边除以

540

540 x = 666 0^{\circ} + 6480 0^{\circ} n - 2340

两边除以

540

\frac{540 x}{540} = 12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

化简

\frac{540 x}{540} = 12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

化简

\frac{540 x}{540} : x

\frac{540 x}{540}

数字相除:

\frac{540}{540} = 1

= x

化简

12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540} : 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

12.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

对同类项分组

= 12.33333 \dots^{\circ} - \frac{2340}{540} + \frac{6480 0 ^{\circ} n}{540}

消掉

\frac{2340}{540} : \frac{13}{3}

\frac{2340}{540}

约分:

180

= \frac{13}{3}

= 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{6480 0 ^{\circ} n}{540}

消掉

\frac{6480 0 ^{\circ} n}{540} : \frac{36 0 ^{\circ} n}{3}

\frac{6480 0 ^{\circ} n}{540}

约分:

180

= \frac{36 0 ^{\circ} n}{3}

= 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

解

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n : x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

将

13

到右边

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n

两边减去

13

13 + 2 x + \frac{180 x + 954 0 ^{\circ}}{180} - 13 = 27 0^{\circ} + 36 0^{\circ} n - 13

化简

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n - 13

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n - 13

在两边乘以

180

2 x + \frac{180 x + 954 0 ^{\circ}}{180} = 27 0^{\circ} + 36 0^{\circ} n - 13

在两边乘以

180

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 27 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

化简

2 x \cdot 180 + \frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 = 27 0^{\circ} \cdot 180 + 36 0^{\circ} n \cdot 180 - 13 \cdot 180

化简

2 x \cdot 180 : 360 x

2 x \cdot 180

数字相乘:

2 \cdot 180 = 360

= 360 x

化简

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180 : 180 x + 954 0^{\circ}

\frac{180 x + 954 0 ^{\circ}}{180} \cdot 180

分式相乘:

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

= \frac{( 180 x + 954 0 ^{\circ} ) \cdot 180}{180}

约分:

180

= 180 x + 954 0^{\circ}

化简

27 0^{\circ} \cdot 180 : 4860 0^{\circ}

27 0^{\circ} \cdot 180

分式相乘:

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

= 4860 0^{\circ}

数字相乘:

3 \cdot 180 = 540

= 4860 0^{\circ}

数字相除:

\frac{540}{2} = 270

= 4860 0^{\circ}

化简

36 0^{\circ} n \cdot 180 : 6480 0^{\circ} n

36 0^{\circ} n \cdot 180

数字相乘:

2 \cdot 180 = 360

= 6480 0^{\circ} n

化简

- 13 \cdot 180 : - 2340

- 13 \cdot 180

数字相乘:

13 \cdot 180 = 2340

= - 2340

360 x + 180 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

将

954 0^{\circ}

到右边

540 x + 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340

两边减去

954 0^{\circ}

540 x + 954 0^{\circ} - 954 0^{\circ} = 4860 0^{\circ} + 6480 0^{\circ} n - 2340 - 954 0^{\circ}

化简

540 x = 3906 0^{\circ} + 6480 0^{\circ} n - 2340

540 x = 3906 0^{\circ} + 6480 0^{\circ} n - 2340

两边除以

540

540 x = 3906 0^{\circ} + 6480 0^{\circ} n - 2340

两边除以

540

\frac{540 x}{540} = 72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

化简

\frac{540 x}{540} = 72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

化简

\frac{540 x}{540} : x

\frac{540 x}{540}

数字相除:

\frac{540}{540} = 1

= x

化简

72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540} : 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

72.33333 \dots^{\circ} + \frac{6480 0 ^{\circ} n}{540} - \frac{2340}{540}

对同类项分组

= 72.33333 \dots^{\circ} - \frac{2340}{540} + \frac{6480 0 ^{\circ} n}{540}

消掉

\frac{2340}{540} : \frac{13}{3}

\frac{2340}{540}

约分:

180

= \frac{13}{3}

= 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{6480 0 ^{\circ} n}{540}

消掉

\frac{6480 0 ^{\circ} n}{540} : \frac{36 0 ^{\circ} n}{3}

\frac{6480 0 ^{\circ} n}{540}

约分:

180

= \frac{36 0 ^{\circ} n}{3}

= 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

x = 12.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}, x = 72.33333 \dots^{\circ} - \frac{13}{3} + \frac{36 0 ^{\circ} n}{3}

作图

流行的例子

sin(2x)-sqrt(3)sin(x)=0

sin (2 x) - 3 sin (x) = 0

2sin(2x)+sqrt(3)cos(x)=0

2 sin (2 x) + 3 cos (x) = 0

-6sqrt(3)cot(θ)+1=-5

- 63 cot (θ) + 1 = - 5

5 - 10 sin (2 x) = 0

solvefor x,cos(3x)=0

so l v e f or x, cos (3 x) = 0