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受欢迎的三角函数 >

tan(2x+1)=-cot(x+3)

解答

tan (2 x + 1) = - cot (x + 3)

解答

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

+1

度数

x = 204.59155 \dots^{\circ} + 36 0^{\circ} n, x = 384.59155 \dots^{\circ} + 36 0^{\circ} n

求解步骤

tan (2 x + 1) = - cot (x + 3)

两边减去

- cot (x + 3)

tan (2 x + 1) + cot (x + 3) = 0

用 sin, cos 表示

cot (3 + x) + tan (1 + 2 x)

使用基本三角恒等式:

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

= \frac{cos ( 3 + x )}{sin ( 3 + x )} + tan (1 + 2 x)

使用基本三角恒等式:

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

= \frac{cos ( 3 + x )}{sin ( 3 + x )} + \frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )}

化简

\frac{cos ( 3 + x )}{sin ( 3 + x )} + \frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )} : \frac{cos ( 3 + x ) cos ( 2 x + 1 ) + sin ( 1 + 2 x ) sin ( x + 3 )}{sin ( x + 3 ) cos ( 2 x + 1 )}

\frac{cos ( 3 + x )}{sin ( 3 + x )} + \frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )}

sin (3 + x), cos (1 + 2 x)

的最小公倍数:

sin (x + 3) cos (2 x + 1)

sin (3 + x), cos (1 + 2 x)

最小公倍数 (LCM)

计算出由出现在

sin (3 + x)

或

cos (1 + 2 x)

中的因子组成的表达式

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

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 + 3) cos (2 x + 1)

对于

\frac{cos ( 3 + x )}{sin ( 3 + x )} :

将分母和分子乘以

cos (2 x + 1)

\frac{cos ( 3 + x )}{sin ( 3 + x )} = \frac{cos ( 3 + x ) cos ( 2 x + 1 )}{sin ( 3 + x ) cos ( 2 x + 1 )}

对于

\frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )} :

将分母和分子乘以

sin (x + 3)

\frac{sin ( 1 + 2 x )}{cos ( 1 + 2 x )} = \frac{sin ( 1 + 2 x ) sin ( x + 3 )}{cos ( 1 + 2 x ) sin ( x + 3 )}

= \frac{cos ( 3 + x ) cos ( 2 x + 1 )}{sin ( 3 + x ) cos ( 2 x + 1 )} + \frac{sin ( 1 + 2 x ) sin ( x + 3 )}{cos ( 1 + 2 x ) sin ( x + 3 )}

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 ( 3 + x ) cos ( 2 x + 1 ) + sin ( 1 + 2 x ) sin ( x + 3 )}{sin ( x + 3 ) cos ( 2 x + 1 )}

= \frac{cos ( 3 + x ) cos ( 2 x + 1 ) + sin ( 1 + 2 x ) sin ( x + 3 )}{sin ( x + 3 ) cos ( 2 x + 1 )}

\frac{cos ( 1 + 2 x ) cos ( 3 + x ) + sin ( 1 + 2 x ) sin ( 3 + x )}{cos ( 1 + 2 x ) sin ( 3 + x )} = 0

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

cos (1 + 2 x) cos (3 + x) + sin (1 + 2 x) sin (3 + x) = 0

使用三角恒等式改写

cos (1 + 2 x) cos (3 + x) + sin (1 + 2 x) sin (3 + x)

使用角差恒等式:

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

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

cos (1 + 2 x - (3 + x)) = 0

cos (1 + 2 x - (3 + 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}

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn, 1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn, 1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

解

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn : x = 2 πn + 2 + \frac{π}{2}

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn

将

1

到右边

1 + 2 x - (3 + x) = \frac{π}{2} + 2 πn

两边减去

1

1 + 2 x - (3 + x) - 1 = \frac{π}{2} + 2 πn - 1

化简

2 x - (3 + x) = \frac{π}{2} + 2 πn - 1

2 x - (3 + x) = \frac{π}{2} + 2 πn - 1

展开

2 x - (3 + x) : x - 3

2 x - (3 + x)

- (3 + x) : - 3 - x

- (3 + x)

打开括号

= - (3) - (x)

使用加减运算法则

+ (- a) = - a

= - 3 - x

= 2 x - 3 - x

化简

2 x - 3 - x : x - 3

2 x - 3 - x

对同类项分组

= 2 x - x - 3

同类项相加:

2 x - x = x

= x - 3

= x - 3

x - 3 = \frac{π}{2} + 2 πn - 1

将

3

到右边

x - 3 = \frac{π}{2} + 2 πn - 1

两边加上

3

x - 3 + 3 = \frac{π}{2} + 2 πn - 1 + 3

化简

x - 3 + 3 = \frac{π}{2} + 2 πn - 1 + 3

化简

x - 3 + 3 : x

x - 3 + 3

同类项相加:

- 3 + 3 = 0

= x

化简

\frac{π}{2} + 2 πn - 1 + 3 : 2 πn + 2 + \frac{π}{2}

\frac{π}{2} + 2 πn - 1 + 3

数字相加/相减:

- 1 + 3 = 2

= 2 πn + 2 + \frac{π}{2}

x = 2 πn + 2 + \frac{π}{2}

x = 2 πn + 2 + \frac{π}{2}

x = 2 πn + 2 + \frac{π}{2}

解

1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn : x = 2 πn + 2 + \frac{3 π}{2}

1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

将

1

到右边

1 + 2 x - (3 + x) = \frac{3 π}{2} + 2 πn

两边减去

1

1 + 2 x - (3 + x) - 1 = \frac{3 π}{2} + 2 πn - 1

化简

2 x - (3 + x) = \frac{3 π}{2} + 2 πn - 1

2 x - (3 + x) = \frac{3 π}{2} + 2 πn - 1

展开

2 x - (3 + x) : x - 3

2 x - (3 + x)

- (3 + x) : - 3 - x

- (3 + x)

打开括号

= - (3) - (x)

使用加减运算法则

+ (- a) = - a

= - 3 - x

= 2 x - 3 - x

化简

2 x - 3 - x : x - 3

2 x - 3 - x

对同类项分组

= 2 x - x - 3

同类项相加:

2 x - x = x

= x - 3

= x - 3

x - 3 = \frac{3 π}{2} + 2 πn - 1

将

3

到右边

x - 3 = \frac{3 π}{2} + 2 πn - 1

两边加上

3

x - 3 + 3 = \frac{3 π}{2} + 2 πn - 1 + 3

化简

x - 3 + 3 = \frac{3 π}{2} + 2 πn - 1 + 3

化简

x - 3 + 3 : x

x - 3 + 3

同类项相加:

- 3 + 3 = 0

= x

化简

\frac{3 π}{2} + 2 πn - 1 + 3 : 2 πn + 2 + \frac{3 π}{2}

\frac{3 π}{2} + 2 πn - 1 + 3

数字相加/相减:

- 1 + 3 = 2

= 2 πn + 2 + \frac{3 π}{2}

x = 2 πn + 2 + \frac{3 π}{2}

x = 2 πn + 2 + \frac{3 π}{2}

x = 2 πn + 2 + \frac{3 π}{2}

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

作图

流行的例子

(4sec^2(x))/2 =-8sec(x)

\frac{4 sec ^{2} ( x )}{2} = - 8 sec (x)

tan(x)+tan^3(x)=0

tan (x) + tan^{3} (x) = 0

sin^3(x)-2sin(x)=4cos^2(x)-3

sin^{3} (x) - 2 sin (x) = 4 cos^{2} (x) - 3

cos^2(x)+6sin(x)-6=0

cos^{2} (x) + 6 sin (x) - 6 = 0

(1-sin(x))*cos^3(x)=0

(1 - sin (x)) \cdot cos^{3} (x) = 0