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受欢迎的三角函数 >

4cot(x)=cot(x)sin^2(x)

解答

4 cot (x) = cot (x) sin^{2} (x)

解答

x = \frac{π}{2} + πn

+1

度数

x = 9 0^{\circ} + 18 0^{\circ} n

求解步骤

4 cot (x) = cot (x) sin^{2} (x)

两边减去

cot (x) sin^{2} (x)

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

分解

4 cot (x) - sin^{2} (x) cot (x) : - cot (x) (sin (x) + 2) (sin (x) - 2)

4 cot (x) - sin^{2} (x) cot (x)

因式分解出通项

- cot (x)

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

分解

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

sin^{2} (x) - 4

将

4

改写为

2^{2}

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

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

分别求解每个部分

cot (x) = 0 or sin (x) + 2 = 0 or sin (x) - 2 = 0

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = 0

cot (x) = 0

的通解

cot (x)

周期表（周期为

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

sin (x) + 2 = 0 :

无解

sin (x) + 2 = 0

将

2

到右边

sin (x) + 2 = 0

两边减去

2

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

化简

sin (x) = - 2

sin (x) = - 2

- 1 \leq sin (x) \leq 1

无解

sin (x) - 2 = 0 :

无解

sin (x) - 2 = 0

将

2

到右边

sin (x) - 2 = 0

两边加上

2

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

化简

sin (x) = 2

sin (x) = 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = \frac{π}{2} + πn

作图

流行的例子

arctan (x) = - 1

9cos(x)tan(x)=-4tan(x)

9 cos (x) tan (x) = - 4 tan (x)

cos(θ)-sin(θ)=0

cos (θ) - sin (θ) = 0

tan^2(x)-9tan(x)-10=0

tan^{2} (x) - 9 tan (x) - 10 = 0

(cot(θ)-1)(csc(θ)-1)=0

(cot (θ) - 1) (csc (θ) - 1) = 0