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受欢迎的三角函数 >

tan(5x)=cot(x)

解答

tan (5 x) = cot (x)

解答

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

+1

度数

x = 1 5^{\circ} + 6 0^{\circ} n, x = 4 5^{\circ} + 6 0^{\circ} n

求解步骤

tan (5 x) = cot (x)

两边减去

cot (x)

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

用 sin, cos 表示

- cot (x) + tan (5 x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

sin (x), cos (5 x)

的最小公倍数:

sin (x) cos (5 x)

sin (x), cos (5 x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (5 x)

中的因子组成的表达式

= sin (x) cos (5 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 (x) cos (5 x)

对于

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

将分母和分子乘以

cos (5 x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

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

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

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

使用三角恒等式改写

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

- cos (5 x + x) = 0

两边除以

- 1

- cos (5 x + x) = 0

两边除以

- 1

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

化简

cos (5 x + x) = 0

cos (5 x + x) = 0

cos (5 x + 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}

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

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

解

5 x + x = \frac{π}{2} + 2 πn : x = \frac{π}{12} + \frac{πn}{3}

5 x + x = \frac{π}{2} + 2 πn

同类项相加:

5 x + x = 6 x

6 x = \frac{π}{2} + 2 πn

两边除以

6

6 x = \frac{π}{2} + 2 πn

两边除以

6

\frac{6 x}{6} = \frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} = \frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} : x

\frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{π}{2}}{6} + \frac{2 πn}{6} : \frac{π}{12} + \frac{πn}{3}

\frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

\frac{\frac{π}{2}}{6} = \frac{π}{12}

\frac{\frac{π}{2}}{6}

使用分式法则:

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

= \frac{π}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{π}{12}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

约分:

2

= \frac{πn}{3}

= \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

解

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

5 x + x = \frac{3 π}{2} + 2 πn

同类项相加:

5 x + x = 6 x

6 x = \frac{3 π}{2} + 2 πn

两边除以

6

6 x = \frac{3 π}{2} + 2 πn

两边除以

6

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

化简

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

化简

\frac{6 x}{6} : x

\frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{3 π}{2}}{6} + \frac{2 πn}{6} : \frac{π}{4} + \frac{πn}{3}

\frac{\frac{3 π}{2}}{6} + \frac{2 πn}{6}

\frac{\frac{3 π}{2}}{6} = \frac{π}{4}

\frac{\frac{3 π}{2}}{6}

使用分式法则:

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

= \frac{3 π}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{3 π}{12}

约分:

3

= \frac{π}{4}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

约分:

2

= \frac{πn}{3}

= \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

作图

流行的例子

1/4 sin(8x)=-1/8 ,2cos(8x)=sqrt(3)

\frac{1}{4} sin (8 x) = - \frac{1}{8}, 2 cos (8 x) = 3

sin(x)-sin(2x)=0,0<= x<2pi

sin (x) - sin (2 x) = 0, 0 \leq x < 2 π

sin(2x)=(-2sqrt(6))/7

sin (2 x) = \frac{- 2 6}{7}

arctan(4x)+arctan(6x)=90

arctan (4 x) + arctan (6 x) = 90

2sin(θ)tan(θ)-tan(θ)=1-2sin(θ)

2 sin (θ) tan (θ) - tan (θ) = 1 - 2 sin (θ)