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受欢迎的三角函数 >

cot(x)cos^2(x)=cot(x)

解答

cot (x) cos^{2} (x) = cot (x)

解答

x = \frac{π}{2} + πn

+1

度数

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

求解步骤

cot (x) cos^{2} (x) = cot (x)

两边减去

cot (x)

cos^{2} (x) cot (x) - cot (x) = 0

分解

cos^{2} (x) cot (x) - cot (x) : cot (x) (cos (x) + 1) (cos (x) - 1)

cos^{2} (x) cot (x) - cot (x)

因式分解出通项

cot (x)

= cot (x) (cos^{2} (x) - 1)

分解

cos^{2} (x) - 1 : (cos (x) + 1) (cos (x) - 1)

cos^{2} (x) - 1

将

1

改写为

1^{2}

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

使用平方差公式:

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

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

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

= cot (x) (cos (x) + 1) (cos (x) - 1)

cot (x) (cos (x) + 1) (cos (x) - 1) = 0

分别求解每个部分

cot (x) = 0 or cos (x) + 1 = 0 or cos (x) - 1 = 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

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

将

1

到右边

cos (x) + 1 = 0

两边减去

1

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

化简

cos (x) = - 1

cos (x) = - 1

cos (x) = - 1

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}

x = π + 2 πn

x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

将

1

到右边

cos (x) - 1 = 0

两边加上

1

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

化简

cos (x) = 1

cos (x) = 1

cos (x) = 1

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}

x = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

合并所有解

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

因为方程对以下值无定义:

π + 2 πn, 2 πn

x = \frac{π}{2} + πn

作图

流行的例子

2 sin (2 x) + 5 = 4

solvefor x,sin(x)=cos(x)

so l v e f or x, sin (x) = cos (x)

sin(θ)=-(sqrt(3))/2 ,0<= θ<= 2pi

sin (θ) = - \frac{3}{2}, 0 \leq θ \leq 2 π

sin(x)+cos(x)= 1/(sqrt(2))

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

cot (x) = 4