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受欢迎的三角函数 >

tan(8b)=cot(10b)

解答

tan (8 b) = cot (10 b)

解答

b = \frac{π}{36} + \frac{πn}{9}, b = \frac{π}{12} + \frac{πn}{9}

+1

度数

b = 5^{\circ} + 2 0^{\circ} n, b = 1 5^{\circ} + 2 0^{\circ} n

求解步骤

tan (8 b) = cot (10 b)

两边减去

cot (10 b)

tan (8 b) - cot (10 b) = 0

用 sin, cos 表示

- cot (10 b) + tan (8 b)

使用基本三角恒等式:

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

= - \frac{cos ( 10 b )}{sin ( 10 b )} + tan (8 b)

使用基本三角恒等式:

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

= - \frac{cos ( 10 b )}{sin ( 10 b )} + \frac{sin ( 8 b )}{cos ( 8 b )}

化简

- \frac{cos ( 10 b )}{sin ( 10 b )} + \frac{sin ( 8 b )}{cos ( 8 b )} : \frac{- cos ( 10 b ) cos ( 8 b ) + sin ( 8 b ) sin ( 10 b )}{sin ( 10 b ) cos ( 8 b )}

- \frac{cos ( 10 b )}{sin ( 10 b )} + \frac{sin ( 8 b )}{cos ( 8 b )}

sin (10 b), cos (8 b)

的最小公倍数:

sin (10 b) cos (8 b)

sin (10 b), cos (8 b)

最小公倍数 (LCM)

计算出由出现在

sin (10 b)

或

cos (8 b)

中的因子组成的表达式

= sin (10 b) cos (8 b)

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 (10 b) cos (8 b)

对于

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

将分母和分子乘以

cos (8 b)

\frac{cos ( 10 b )}{sin ( 10 b )} = \frac{cos ( 10 b ) cos ( 8 b )}{sin ( 10 b ) cos ( 8 b )}

对于

\frac{sin ( 8 b )}{cos ( 8 b )} :

将分母和分子乘以

sin (10 b)

\frac{sin ( 8 b )}{cos ( 8 b )} = \frac{sin ( 8 b ) sin ( 10 b )}{cos ( 8 b ) sin ( 10 b )}

= - \frac{cos ( 10 b ) cos ( 8 b )}{sin ( 10 b ) cos ( 8 b )} + \frac{sin ( 8 b ) sin ( 10 b )}{cos ( 8 b ) sin ( 10 b )}

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 ( 10 b ) cos ( 8 b ) + sin ( 8 b ) sin ( 10 b )}{sin ( 10 b ) cos ( 8 b )}

= \frac{- cos ( 10 b ) cos ( 8 b ) + sin ( 8 b ) sin ( 10 b )}{sin ( 10 b ) cos ( 8 b )}

\frac{- cos ( 10 b ) cos ( 8 b ) + sin ( 10 b ) sin ( 8 b )}{cos ( 8 b ) sin ( 10 b )} = 0

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

- cos (10 b) cos (8 b) + sin (10 b) sin (8 b) = 0

使用三角恒等式改写

- cos (10 b) cos (8 b) + sin (10 b) sin (8 b)

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 b + 8 b)

- cos (10 b + 8 b) = 0

两边除以

- 1

- cos (10 b + 8 b) = 0

两边除以

- 1

\frac{- cos ( 10 b + 8 b )}{- 1} = \frac{0}{- 1}

化简

cos (10 b + 8 b) = 0

cos (10 b + 8 b) = 0

cos (10 b + 8 b) = 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 b + 8 b = \frac{π}{2} + 2 πn, 10 b + 8 b = \frac{3 π}{2} + 2 πn

10 b + 8 b = \frac{π}{2} + 2 πn, 10 b + 8 b = \frac{3 π}{2} + 2 πn

解

10 b + 8 b = \frac{π}{2} + 2 πn : b = \frac{π}{36} + \frac{πn}{9}

10 b + 8 b = \frac{π}{2} + 2 πn

同类项相加:

10 b + 8 b = 18 b

18 b = \frac{π}{2} + 2 πn

两边除以

18

18 b = \frac{π}{2} + 2 πn

两边除以

18

\frac{18 b}{18} = \frac{\frac{π}{2}}{18} + \frac{2 πn}{18}

化简

\frac{18 b}{18} = \frac{\frac{π}{2}}{18} + \frac{2 πn}{18}

化简

\frac{18 b}{18} : b

\frac{18 b}{18}

数字相除:

\frac{18}{18} = 1

= b

化简

\frac{\frac{π}{2}}{18} + \frac{2 πn}{18} : \frac{π}{36} + \frac{πn}{9}

\frac{\frac{π}{2}}{18} + \frac{2 πn}{18}

\frac{\frac{π}{2}}{18} = \frac{π}{36}

\frac{\frac{π}{2}}{18}

使用分式法则:

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

= \frac{π}{2 \cdot 18}

数字相乘:

2 \cdot 18 = 36

= \frac{π}{36}

\frac{2 πn}{18} = \frac{πn}{9}

\frac{2 πn}{18}

约分:

2

= \frac{πn}{9}

= \frac{π}{36} + \frac{πn}{9}

b = \frac{π}{36} + \frac{πn}{9}

b = \frac{π}{36} + \frac{πn}{9}

b = \frac{π}{36} + \frac{πn}{9}

解

10 b + 8 b = \frac{3 π}{2} + 2 πn : b = \frac{π}{12} + \frac{πn}{9}

10 b + 8 b = \frac{3 π}{2} + 2 πn

同类项相加:

10 b + 8 b = 18 b

18 b = \frac{3 π}{2} + 2 πn

两边除以

18

18 b = \frac{3 π}{2} + 2 πn

两边除以

18

\frac{18 b}{18} = \frac{\frac{3 π}{2}}{18} + \frac{2 πn}{18}

化简

\frac{18 b}{18} = \frac{\frac{3 π}{2}}{18} + \frac{2 πn}{18}

化简

\frac{18 b}{18} : b

\frac{18 b}{18}

数字相除:

\frac{18}{18} = 1

= b

化简

\frac{\frac{3 π}{2}}{18} + \frac{2 πn}{18} : \frac{π}{12} + \frac{πn}{9}

\frac{\frac{3 π}{2}}{18} + \frac{2 πn}{18}

\frac{\frac{3 π}{2}}{18} = \frac{π}{12}

\frac{\frac{3 π}{2}}{18}

使用分式法则:

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

= \frac{3 π}{2 \cdot 18}

数字相乘:

2 \cdot 18 = 36

= \frac{3 π}{36}

约分:

3

= \frac{π}{12}

\frac{2 πn}{18} = \frac{πn}{9}

\frac{2 πn}{18}

约分:

2

= \frac{πn}{9}

= \frac{π}{12} + \frac{πn}{9}

b = \frac{π}{12} + \frac{πn}{9}

b = \frac{π}{12} + \frac{πn}{9}

b = \frac{π}{12} + \frac{πn}{9}

b = \frac{π}{36} + \frac{πn}{9}, b = \frac{π}{12} + \frac{πn}{9}

作图

流行的例子

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