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受欢迎的三角函数 >

tan(x)+cot(x)=4sin(2x)

解答

tan (x) + cot (x) = 4 sin (2 x)

解答

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn, x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

+1

度数

x = 22. 5^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n, x = 112. 5^{\circ} + 18 0^{\circ} n, x = 157. 5^{\circ} + 18 0^{\circ} n

求解步骤

tan (x) + cot (x) = 4 sin (2 x)

两边减去

4 sin (2 x)

tan (x) + cot (x) - 4 sin (2 x) = 0

用 sin, cos 表示

cot (x) + tan (x) - 4 sin (2 x)

使用基本三角恒等式:

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

= \frac{cos ( x )}{sin ( x )} + tan (x) - 4 sin (2 x)

使用基本三角恒等式:

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

= \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )} - 4 sin (2 x)

化简

\frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )} - 4 sin (2 x) : \frac{cos ^{2} ( x ) + sin ^{2} ( x ) - 4 sin ( 2 x ) sin ( x ) cos ( x )}{sin ( x ) cos ( x )}

\frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )} - 4 sin (2 x)

将项转换为分式:

4 sin (2 x) = \frac{4 sin ( 2 x )}{1}

= \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )} - \frac{4 sin ( 2 x )}{1}

sin (x), cos (x), 1

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x), 1

最小公倍数 (LCM)

计算出由至少在以下一个因式表达式中出现的因子组成的表达式

= sin (x) 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

sin (x) cos (x)

对于

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

将分母和分子乘以

cos (x)

\frac{cos ( x )}{sin ( x )} = \frac{cos ( x ) cos ( x )}{sin ( x ) cos ( x )} = \frac{cos ^{2} ( x )}{sin ( x ) cos ( x )}

对于

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

将分母和分子乘以

sin (x)

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x ) sin ( x )}{cos ( x ) sin ( x )} = \frac{sin ^{2} ( x )}{sin ( x ) cos ( x )}

对于

\frac{4 sin ( 2 x )}{1} :

将分母和分子乘以

sin (x) cos (x)

\frac{4 sin ( 2 x )}{1} = \frac{4 sin ( 2 x ) sin ( x ) cos ( x )}{1 \cdot sin ( x ) cos ( x )} = \frac{4 sin ( 2 x ) sin ( x ) cos ( x )}{sin ( x ) cos ( x )}

= \frac{cos ^{2} ( x )}{sin ( x ) cos ( x )} + \frac{sin ^{2} ( x )}{sin ( x ) cos ( x )} - \frac{4 sin ( 2 x ) sin ( x ) cos ( x )}{sin ( x ) cos ( 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 ^{2} ( x ) + sin ^{2} ( x ) - 4 sin ( 2 x ) sin ( x ) cos ( x )}{sin ( x ) cos ( x )}

= \frac{cos ^{2} ( x ) + sin ^{2} ( x ) - 4 sin ( 2 x ) sin ( x ) cos ( x )}{sin ( x ) cos ( x )}

\frac{cos ^{2} ( x ) + sin ^{2} ( x ) - 4 cos ( x ) sin ( 2 x ) sin ( x )}{cos ( x ) sin ( x )} = 0

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

cos^{2} (x) + sin^{2} (x) - 4 cos (x) sin (2 x) sin (x) = 0

使用三角恒等式改写

cos^{2} (x) + sin^{2} (x) - 4 cos (x) sin (2 x) sin (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

= - 4 cos (x) sin (2 x) sin (x) + 1

使用倍角公式:

2 sin (x) cos (x) = sin (2 x)

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

= 1 - 4 \cdot \frac{sin ( 2 x )}{2} sin (2 x)

4 \cdot \frac{sin ( 2 x )}{2} sin (2 x) = 2 sin^{2} (2 x)

4 \cdot \frac{sin ( 2 x )}{2} sin (2 x)

分式相乘:

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

= \frac{sin ( 2 x ) \cdot 4 sin ( 2 x )}{2}

sin (2 x) \cdot 4 sin (2 x) = 4 sin^{2} (2 x)

sin (2 x) \cdot 4 sin (2 x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (2 x) sin (2 x) = sin^{1 + 1} (2 x)

= 4 sin^{1 + 1} (2 x)

数字相加:

1 + 1 = 2

= 4 sin^{2} (2 x)

= \frac{4 sin ^{2} ( 2 x )}{2}

数字相除:

\frac{4}{2} = 2

= 2 sin^{2} (2 x)

= 1 - 2 sin^{2} (2 x)

1 - 2 sin^{2} (2 x) = 0

用替代法求解

1 - 2 sin^{2} (2 x) = 0

令

: sin (2 x) = u

1 - 2 u^{2} = 0

1 - 2 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

1 - 2 u^{2} = 0

将

1

到右边

1 - 2 u^{2} = 0

两边减去

1

1 - 2 u^{2} - 1 = 0 - 1

化简

- 2 u^{2} = - 1

- 2 u^{2} = - 1

两边除以

- 2

- 2 u^{2} = - 1

两边除以

- 2

\frac{- 2 u ^{2}}{- 2} = \frac{- 1}{- 2}

化简

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{1}{2}, u = - \frac{1}{2}

u = sin (2 x)

代回

sin (2 x) = \frac{1}{2}, sin (2 x) = - \frac{1}{2}

sin (2 x) = \frac{1}{2}, sin (2 x) = - \frac{1}{2}

sin (2 x) = \frac{1}{2} : x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

sin (2 x) = \frac{1}{2}

sin (2 x) = \frac{1}{2}

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}

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

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

解

2 x = \frac{π}{4} + 2 πn : x = \frac{π}{8} + πn

2 x = \frac{π}{4} + 2 πn

两边除以

2

2 x = \frac{π}{4} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

使用分式法则:

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

= \frac{π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

解

2 x = \frac{3 π}{4} + 2 πn : x = \frac{3 π}{8} + πn

2 x = \frac{3 π}{4} + 2 πn

两边除以

2

2 x = \frac{3 π}{4} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2} : \frac{3 π}{8} + πn

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

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

使用分式法则:

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

= \frac{3 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{3 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

sin (2 x) = - \frac{1}{2} : x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

sin (2 x) = - \frac{1}{2}

sin (2 x) = - \frac{1}{2}

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}

2 x = \frac{5 π}{4} + 2 πn, 2 x = \frac{7 π}{4} + 2 πn

2 x = \frac{5 π}{4} + 2 πn, 2 x = \frac{7 π}{4} + 2 πn

解

2 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{8} + πn

2 x = \frac{5 π}{4} + 2 πn

两边除以

2

2 x = \frac{5 π}{4} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2} : \frac{5 π}{8} + πn

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{4}}{2} = \frac{5 π}{8}

\frac{\frac{5 π}{4}}{2}

使用分式法则:

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

= \frac{5 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{5 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

解

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{8} + πn

2 x = \frac{7 π}{4} + 2 πn

两边除以

2

2 x = \frac{7 π}{4} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π}{8} + πn

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{4}}{2} = \frac{7 π}{8}

\frac{\frac{7 π}{4}}{2}

使用分式法则:

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

= \frac{7 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{7 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

合并所有解

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn, x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

作图

流行的例子

tan(θ)=(-sqrt(3))/3

tan (θ) = \frac{- 3}{3}

5cos^3(x)=5cos(x)

5 cos^{3} (x) = 5 cos (x)

sin(2x-pi/4)=(sqrt(2))/2

sin (2 x - \frac{π}{4}) = \frac{2}{2}

tan (2 x) = cot (x)

cos (x) = 0.7