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受欢迎的三角函数 >

tan(x)=tan(2x-30)

解答

tan (x) = tan (2 x - 30)

解答

x = - 2 πn + 30, x = - π + 30 - 2 πn

+1

度数

x = 1718.87338 \dots^{\circ} - 36 0^{\circ} n, x = 1538.87338 \dots^{\circ} - 36 0^{\circ} n

求解步骤

tan (x) = tan (2 x - 30)

两边减去

tan (2 x - 30)

tan (x) - tan (2 x - 30) = 0

用 sin, cos 表示

- tan (- 30 + 2 x) + tan (x)

使用基本三角恒等式:

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

= - \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + tan (x)

使用基本三角恒等式:

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

= - \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + \frac{sin ( x )}{cos ( x )}

化简

- \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + \frac{sin ( x )}{cos ( x )} : \frac{- sin ( - 30 + 2 x ) cos ( x ) + sin ( x ) cos ( 2 x - 30 )}{cos ( 2 x - 30 ) cos ( x )}

- \frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} + \frac{sin ( x )}{cos ( x )}

cos (- 30 + 2 x), cos (x)

的最小公倍数:

cos (2 x - 30) cos (x)

cos (- 30 + 2 x), cos (x)

最小公倍数 (LCM)

计算出由出现在

cos (- 30 + 2 x)

或

cos (x)

中的因子组成的表达式

= cos (2 x - 30) cos (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

cos (2 x - 30) cos (x)

对于

\frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} :

将分母和分子乘以

cos (x)

\frac{sin ( - 30 + 2 x )}{cos ( - 30 + 2 x )} = \frac{sin ( - 30 + 2 x ) cos ( x )}{cos ( - 30 + 2 x ) cos ( x )}

对于

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

将分母和分子乘以

cos (2 x - 30)

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x ) cos ( 2 x - 30 )}{cos ( x ) cos ( 2 x - 30 )}

= - \frac{sin ( - 30 + 2 x ) cos ( x )}{cos ( - 30 + 2 x ) cos ( x )} + \frac{sin ( x ) cos ( 2 x - 30 )}{cos ( x ) cos ( 2 x - 30 )}

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{- sin ( - 30 + 2 x ) cos ( x ) + sin ( x ) cos ( 2 x - 30 )}{cos ( 2 x - 30 ) cos ( x )}

= \frac{- sin ( - 30 + 2 x ) cos ( x ) + sin ( x ) cos ( 2 x - 30 )}{cos ( 2 x - 30 ) cos ( x )}

\frac{cos ( - 30 + 2 x ) sin ( x ) - cos ( x ) sin ( - 30 + 2 x )}{cos ( - 30 + 2 x ) cos ( x )} = 0

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

cos (- 30 + 2 x) sin (x) - cos (x) sin (- 30 + 2 x) = 0

使用三角恒等式改写

cos (- 30 + 2 x) sin (x) - cos (x) sin (- 30 + 2 x)

使用角差恒等式:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (x - (- 30 + 2 x))

sin (x - (- 30 + 2 x)) = 0

sin (x - (- 30 + 2 x)) = 0

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x - (- 30 + 2 x) = 0 + 2 πn, x - (- 30 + 2 x) = π + 2 πn

x - (- 30 + 2 x) = 0 + 2 πn, x - (- 30 + 2 x) = π + 2 πn

解

x - (- 30 + 2 x) = 0 + 2 πn : x = - 2 πn + 30

x - (- 30 + 2 x) = 0 + 2 πn

0 + 2 πn = 2 πn

x - (- 30 + 2 x) = 2 πn

展开

x - (- 30 + 2 x) : - x + 30

x - (- 30 + 2 x)

- (- 30 + 2 x) : 30 - 2 x

- (- 30 + 2 x)

打开括号

= - (- 30) - (2 x)

使用加减运算法则

- (- a) = a, - (a) = - a

= 30 - 2 x

= x + 30 - 2 x

化简

x + 30 - 2 x : - x + 30

x + 30 - 2 x

对同类项分组

= x - 2 x + 30

同类项相加:

x - 2 x = - x

= - x + 30

= - x + 30

- x + 30 = 2 πn

将

30

到右边

- x + 30 = 2 πn

两边减去

30

- x + 30 - 30 = 2 πn - 30

化简

- x = 2 πn - 30

- x = 2 πn - 30

两边除以

- 1

- x = 2 πn - 30

两边除以

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} - \frac{30}{- 1}

化简

\frac{- x}{- 1} = \frac{2 πn}{- 1} - \frac{30}{- 1}

化简

\frac{- x}{- 1} : x

\frac{- x}{- 1}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{x}{1}

使用法则

\frac{a}{1} = a

= x

化简

\frac{2 πn}{- 1} - \frac{30}{- 1} : - 2 πn + 30

\frac{2 πn}{- 1} - \frac{30}{- 1}

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 πn}{1}

使用法则

\frac{a}{1} = a

= - 2 πn

= - 2 πn - \frac{30}{- 1}

\frac{30}{- 1} = - 30

\frac{30}{- 1}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{30}{1}

使用法则

\frac{a}{1} = a

= - 30

= - 2 πn - (- 30)

使用法则

- (- a) = a

= - 2 πn + 30

x = - 2 πn + 30

x = - 2 πn + 30

x = - 2 πn + 30

解

x - (- 30 + 2 x) = π + 2 πn : x = - π + 30 - 2 πn

x - (- 30 + 2 x) = π + 2 πn

展开

x - (- 30 + 2 x) : - x + 30

x - (- 30 + 2 x)

- (- 30 + 2 x) : 30 - 2 x

- (- 30 + 2 x)

打开括号

= - (- 30) - (2 x)

使用加减运算法则

- (- a) = a, - (a) = - a

= 30 - 2 x

= x + 30 - 2 x

化简

x + 30 - 2 x : - x + 30

x + 30 - 2 x

对同类项分组

= x - 2 x + 30

同类项相加:

x - 2 x = - x

= - x + 30

= - x + 30

- x + 30 = π + 2 πn

将

30

到右边

- x + 30 = π + 2 πn

两边减去

30

- x + 30 - 30 = π + 2 πn - 30

化简

- x = π + 2 πn - 30

- x = π + 2 πn - 30

两边除以

- 1

- x = π + 2 πn - 30

两边除以

- 1

\frac{- x}{- 1} = \frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1}

化简

\frac{- x}{- 1} = \frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1}

化简

\frac{- x}{- 1} : x

\frac{- x}{- 1}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{x}{1}

使用法则

\frac{a}{1} = a

= x

化简

\frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1} : - π + 30 - 2 πn

\frac{π}{- 1} + \frac{2 πn}{- 1} - \frac{30}{- 1}

对同类项分组

= \frac{π}{- 1} - \frac{30}{- 1} + \frac{2 πn}{- 1}

\frac{π}{- 1} = - π

\frac{π}{- 1}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{π}{1}

使用法则

\frac{a}{1} = a

= - π

= - π - \frac{30}{- 1} + \frac{2 πn}{- 1}

\frac{30}{- 1} = - 30

\frac{30}{- 1}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{30}{1}

使用法则

\frac{a}{1} = a

= - 30

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 πn}{1}

使用法则

\frac{a}{1} = a

= - 2 πn

= - π - (- 30) - 2 πn

使用法则

- (- a) = a

= - π + 30 - 2 πn

x = - π + 30 - 2 πn

x = - π + 30 - 2 πn

x = - π + 30 - 2 πn

x = - 2 πn + 30, x = - π + 30 - 2 πn

作图

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