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受欢迎的三角函数 >

tan(x+2)=cot(4x+28)

解答

tan (x + 2) = cot (4 x + 28)

解答

x = - 6 + \frac{2 πn}{5} + \frac{π}{10}, x = - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

+1

度数

x = - 325.77467 \dots^{\circ} + 7 2^{\circ} n, x = - 289.77467 \dots^{\circ} + 7 2^{\circ} n

求解步骤

tan (x + 2) = cot (4 x + 28)

两边减去

cot (4 x + 28)

tan (x + 2) - cot (4 x + 28) = 0

用 sin, cos 表示

- cot (28 + 4 x) + tan (2 + x)

使用基本三角恒等式:

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

= - \frac{cos ( 28 + 4 x )}{sin ( 28 + 4 x )} + tan (2 + x)

使用基本三角恒等式:

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

= - \frac{cos ( 28 + 4 x )}{sin ( 28 + 4 x )} + \frac{sin ( 2 + x )}{cos ( 2 + x )}

化简

- \frac{cos ( 28 + 4 x )}{sin ( 28 + 4 x )} + \frac{sin ( 2 + x )}{cos ( 2 + x )} : \frac{- cos ( 28 + 4 x ) cos ( x + 2 ) + sin ( 2 + x ) sin ( 4 x + 28 )}{sin ( 4 x + 28 ) cos ( x + 2 )}

- \frac{cos ( 28 + 4 x )}{sin ( 28 + 4 x )} + \frac{sin ( 2 + x )}{cos ( 2 + x )}

sin (28 + 4 x), cos (2 + x)

的最小公倍数:

sin (4 x + 28) cos (x + 2)

sin (28 + 4 x), cos (2 + x)

最小公倍数 (LCM)

计算出由出现在

sin (28 + 4 x)

或

cos (2 + x)

中的因子组成的表达式

= sin (4 x + 28) cos (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 (4 x + 28) cos (x + 2)

对于

\frac{cos ( 28 + 4 x )}{sin ( 28 + 4 x )} :

将分母和分子乘以

cos (x + 2)

\frac{cos ( 28 + 4 x )}{sin ( 28 + 4 x )} = \frac{cos ( 28 + 4 x ) cos ( x + 2 )}{sin ( 28 + 4 x ) cos ( x + 2 )}

对于

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

将分母和分子乘以

sin (4 x + 28)

\frac{sin ( 2 + x )}{cos ( 2 + x )} = \frac{sin ( 2 + x ) sin ( 4 x + 28 )}{cos ( 2 + x ) sin ( 4 x + 28 )}

= - \frac{cos ( 28 + 4 x ) cos ( x + 2 )}{sin ( 28 + 4 x ) cos ( x + 2 )} + \frac{sin ( 2 + x ) sin ( 4 x + 28 )}{cos ( 2 + x ) sin ( 4 x + 28 )}

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 ( 28 + 4 x ) cos ( x + 2 ) + sin ( 2 + x ) sin ( 4 x + 28 )}{sin ( 4 x + 28 ) cos ( x + 2 )}

= \frac{- cos ( 28 + 4 x ) cos ( x + 2 ) + sin ( 2 + x ) sin ( 4 x + 28 )}{sin ( 4 x + 28 ) cos ( x + 2 )}

\frac{- cos ( 2 + x ) cos ( 28 + 4 x ) + sin ( 2 + x ) sin ( 28 + 4 x )}{cos ( 2 + x ) sin ( 28 + 4 x )} = 0

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

- cos (2 + x) cos (28 + 4 x) + sin (2 + x) sin (28 + 4 x) = 0

使用三角恒等式改写

- cos (2 + x) cos (28 + 4 x) + sin (2 + x) sin (28 + 4 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 (2 + x + 28 + 4 x)

- cos (2 + x + 28 + 4 x) = 0

两边除以

- 1

- cos (2 + x + 28 + 4 x) = 0

两边除以

- 1

\frac{- cos ( 2 + x + 28 + 4 x )}{- 1} = \frac{0}{- 1}

化简

cos (2 + x + 28 + 4 x) = 0

cos (2 + x + 28 + 4 x) = 0

cos (2 + x + 28 + 4 x) = 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}

2 + x + 28 + 4 x = \frac{π}{2} + 2 πn, 2 + x + 28 + 4 x = \frac{3 π}{2} + 2 πn

2 + x + 28 + 4 x = \frac{π}{2} + 2 πn, 2 + x + 28 + 4 x = \frac{3 π}{2} + 2 πn

解

2 + x + 28 + 4 x = \frac{π}{2} + 2 πn : x = - 6 + \frac{2 πn}{5} + \frac{π}{10}

2 + x + 28 + 4 x = \frac{π}{2} + 2 πn

对同类项分组

x + 4 x + 2 + 28 = \frac{π}{2} + 2 πn

同类项相加:

x + 4 x = 5 x

5 x + 2 + 28 = \frac{π}{2} + 2 πn

数字相加:

2 + 28 = 30

5 x + 30 = \frac{π}{2} + 2 πn

将

30

到右边

5 x + 30 = \frac{π}{2} + 2 πn

两边减去

30

5 x + 30 - 30 = \frac{π}{2} + 2 πn - 30

化简

5 x = \frac{π}{2} + 2 πn - 30

5 x = \frac{π}{2} + 2 πn - 30

两边除以

5

5 x = \frac{π}{2} + 2 πn - 30

两边除以

5

\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5}

化简

\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5}

化简

\frac{5 x}{5} : x

\frac{5 x}{5}

数字相除:

\frac{5}{5} = 1

= x

化简

\frac{\frac{π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5} : - 6 + \frac{2 πn}{5} + \frac{π}{10}

\frac{\frac{π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5}

对同类项分组

= - \frac{30}{5} + \frac{2 πn}{5} + \frac{\frac{π}{2}}{5}

\frac{30}{5} = 6

\frac{30}{5}

数字相除:

\frac{30}{5} = 6

= 6

\frac{\frac{π}{2}}{5} = \frac{π}{10}

\frac{\frac{π}{2}}{5}

使用分式法则:

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

= \frac{π}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{π}{10}

= - 6 + \frac{2 πn}{5} + \frac{π}{10}

x = - 6 + \frac{2 πn}{5} + \frac{π}{10}

x = - 6 + \frac{2 πn}{5} + \frac{π}{10}

x = - 6 + \frac{2 πn}{5} + \frac{π}{10}

解

2 + x + 28 + 4 x = \frac{3 π}{2} + 2 πn : x = - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

2 + x + 28 + 4 x = \frac{3 π}{2} + 2 πn

对同类项分组

x + 4 x + 2 + 28 = \frac{3 π}{2} + 2 πn

同类项相加:

x + 4 x = 5 x

5 x + 2 + 28 = \frac{3 π}{2} + 2 πn

数字相加:

2 + 28 = 30

5 x + 30 = \frac{3 π}{2} + 2 πn

将

30

到右边

5 x + 30 = \frac{3 π}{2} + 2 πn

两边减去

30

5 x + 30 - 30 = \frac{3 π}{2} + 2 πn - 30

化简

5 x = \frac{3 π}{2} + 2 πn - 30

5 x = \frac{3 π}{2} + 2 πn - 30

两边除以

5

5 x = \frac{3 π}{2} + 2 πn - 30

两边除以

5

\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5}

化简

\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5}

化简

\frac{5 x}{5} : x

\frac{5 x}{5}

数字相除:

\frac{5}{5} = 1

= x

化简

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5} : - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} - \frac{30}{5}

对同类项分组

= - \frac{30}{5} + \frac{2 πn}{5} + \frac{\frac{3 π}{2}}{5}

\frac{30}{5} = 6

\frac{30}{5}

数字相除:

\frac{30}{5} = 6

= 6

\frac{\frac{3 π}{2}}{5} = \frac{3 π}{10}

\frac{\frac{3 π}{2}}{5}

使用分式法则:

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

= \frac{3 π}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{3 π}{10}

= - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

x = - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

x = - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

x = - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

x = - 6 + \frac{2 πn}{5} + \frac{π}{10}, x = - 6 + \frac{2 πn}{5} + \frac{3 π}{10}

作图

流行的例子

4 cos^{2} (x) - 4 = 0

2 tan (x) + 5 = 3

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

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

3tan^2(x)+10tan(x)+7=0

3 tan^{2} (x) + 10 tan (x) + 7 = 0

8 sin (x) - 8 = 0