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受欢迎的三角函数 >

tan(10x)=cot(5x)

解答

tan (10 x) = cot (5 x)

解答

x = \frac{π}{30} + \frac{2 πn}{15}, x = \frac{π}{10} + \frac{2 πn}{15}

+1

度数

x = 6^{\circ} + 2 4^{\circ} n, x = 1 8^{\circ} + 2 4^{\circ} n

求解步骤

tan (10 x) = cot (5 x)

两边减去

cot (5 x)

tan (10 x) - cot (5 x) = 0

用 sin, cos 表示

- cot (5 x) + tan (10 x)

使用基本三角恒等式:

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

= - \frac{cos ( 5 x )}{sin ( 5 x )} + tan (10 x)

使用基本三角恒等式:

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

= - \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 10 x )}{cos ( 10 x )}

化简

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 10 x )}{cos ( 10 x )} : \frac{- cos ( 5 x ) cos ( 10 x ) + sin ( 10 x ) sin ( 5 x )}{sin ( 5 x ) cos ( 10 x )}

- \frac{cos ( 5 x )}{sin ( 5 x )} + \frac{sin ( 10 x )}{cos ( 10 x )}

sin (5 x), cos (10 x)

的最小公倍数:

sin (5 x) cos (10 x)

sin (5 x), cos (10 x)

最小公倍数 (LCM)

计算出由出现在

sin (5 x)

或

cos (10 x)

中的因子组成的表达式

= sin (5 x) cos (10 x)

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 (5 x) cos (10 x)

对于

\frac{cos ( 5 x )}{sin ( 5 x )} :

将分母和分子乘以

cos (10 x)

\frac{cos ( 5 x )}{sin ( 5 x )} = \frac{cos ( 5 x ) cos ( 10 x )}{sin ( 5 x ) cos ( 10 x )}

对于

\frac{sin ( 10 x )}{cos ( 10 x )} :

将分母和分子乘以

sin (5 x)

\frac{sin ( 10 x )}{cos ( 10 x )} = \frac{sin ( 10 x ) sin ( 5 x )}{cos ( 10 x ) sin ( 5 x )}

= - \frac{cos ( 5 x ) cos ( 10 x )}{sin ( 5 x ) cos ( 10 x )} + \frac{sin ( 10 x ) sin ( 5 x )}{cos ( 10 x ) sin ( 5 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 ( 5 x ) cos ( 10 x ) + sin ( 10 x ) sin ( 5 x )}{sin ( 5 x ) cos ( 10 x )}

= \frac{- cos ( 5 x ) cos ( 10 x ) + sin ( 10 x ) sin ( 5 x )}{sin ( 5 x ) cos ( 10 x )}

\frac{- cos ( 10 x ) cos ( 5 x ) + sin ( 10 x ) sin ( 5 x )}{cos ( 10 x ) sin ( 5 x )} = 0

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

- cos (10 x) cos (5 x) + sin (10 x) sin (5 x) = 0

使用三角恒等式改写

- cos (10 x) cos (5 x) + sin (10 x) sin (5 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 (10 x + 5 x)

- cos (10 x + 5 x) = 0

两边除以

- 1

- cos (10 x + 5 x) = 0

两边除以

- 1

\frac{- cos ( 10 x + 5 x )}{- 1} = \frac{0}{- 1}

化简

cos (10 x + 5 x) = 0

cos (10 x + 5 x) = 0

cos (10 x + 5 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}

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

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

解

10 x + 5 x = \frac{π}{2} + 2 πn : x = \frac{π}{30} + \frac{2 πn}{15}

10 x + 5 x = \frac{π}{2} + 2 πn

同类项相加:

10 x + 5 x = 15 x

15 x = \frac{π}{2} + 2 πn

两边除以

15

15 x = \frac{π}{2} + 2 πn

两边除以

15

\frac{15 x}{15} = \frac{\frac{π}{2}}{15} + \frac{2 πn}{15}

化简

\frac{15 x}{15} = \frac{\frac{π}{2}}{15} + \frac{2 πn}{15}

化简

\frac{15 x}{15} : x

\frac{15 x}{15}

数字相除:

\frac{15}{15} = 1

= x

化简

\frac{\frac{π}{2}}{15} + \frac{2 πn}{15} : \frac{π}{30} + \frac{2 πn}{15}

\frac{\frac{π}{2}}{15} + \frac{2 πn}{15}

\frac{\frac{π}{2}}{15} = \frac{π}{30}

\frac{\frac{π}{2}}{15}

使用分式法则:

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

= \frac{π}{2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{π}{30}

= \frac{π}{30} + \frac{2 πn}{15}

x = \frac{π}{30} + \frac{2 πn}{15}

x = \frac{π}{30} + \frac{2 πn}{15}

x = \frac{π}{30} + \frac{2 πn}{15}

解

10 x + 5 x = \frac{3 π}{2} + 2 πn : x = \frac{π}{10} + \frac{2 πn}{15}

10 x + 5 x = \frac{3 π}{2} + 2 πn

同类项相加:

10 x + 5 x = 15 x

15 x = \frac{3 π}{2} + 2 πn

两边除以

15

15 x = \frac{3 π}{2} + 2 πn

两边除以

15

\frac{15 x}{15} = \frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15}

化简

\frac{15 x}{15} = \frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15}

化简

\frac{15 x}{15} : x

\frac{15 x}{15}

数字相除:

\frac{15}{15} = 1

= x

化简

\frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15} : \frac{π}{10} + \frac{2 πn}{15}

\frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15}

\frac{\frac{3 π}{2}}{15} = \frac{π}{10}

\frac{\frac{3 π}{2}}{15}

使用分式法则:

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

= \frac{3 π}{2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{3 π}{30}

约分:

3

= \frac{π}{10}

= \frac{π}{10} + \frac{2 πn}{15}

x = \frac{π}{10} + \frac{2 πn}{15}

x = \frac{π}{10} + \frac{2 πn}{15}

x = \frac{π}{10} + \frac{2 πn}{15}

x = \frac{π}{30} + \frac{2 πn}{15}, x = \frac{π}{10} + \frac{2 πn}{15}

作图

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