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受欢迎的三角函数 >

sec^2(x)csc(x)-2csc(x)=2-sec^2(x)

解答

sec^{2} (x) csc (x) - 2 csc (x) = 2 - sec^{2} (x)

解答

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

sec^{2} (x) csc (x) - 2 csc (x) = 2 - sec^{2} (x)

两边减去

2 - sec^{2} (x)

sec^{2} (x) csc (x) - 2 csc (x) - 2 + sec^{2} (x) = 0

分解

sec^{2} (x) csc (x) - 2 csc (x) - 2 + sec^{2} (x) : (sec (x) + 2) (sec (x) - 2) (csc (x) + 1)

sec^{2} (x) csc (x) - 2 csc (x) - 2 + sec^{2} (x)

= (sec^{2} (x) csc (x) - 2 csc (x)) + (sec^{2} (x) - 2)

从

sec^{2} (x) csc (x) - 2 csc (x)

分解出因式

csc (x)

:

csc (x) (sec^{2} (x) - 2)

sec^{2} (x) csc (x) - 2 csc (x)

因式分解出通项

csc (x)

= csc (x) (sec^{2} (x) - 2)

= (sec^{2} (x) - 2) + csc (x) (sec^{2} (x) - 2)

因式分解出通项

sec^{2} (x) - 2

= (sec^{2} (x) - 2) (csc (x) + 1)

分解

sec^{2} (x) - 2 : (sec (x) + 2) (sec (x) - 2)

sec^{2} (x) - 2

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用平方差公式:

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

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

= (sec (x) + 2) (sec (x) - 2)

= (sec (x) + 2) (sec (x) - 2) (csc (x) + 1)

(sec (x) + 2) (sec (x) - 2) (csc (x) + 1) = 0

分别求解每个部分

sec (x) + 2 = 0 or sec (x) - 2 = 0 or csc (x) + 1 = 0

sec (x) + 2 = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) + 2 = 0

将

2

到右边

sec (x) + 2 = 0

两边减去

2

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

化简

sec (x) = - 2

sec (x) = - 2

sec (x) = - 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) - 2 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) - 2 = 0

将

2

到右边

sec (x) - 2 = 0

两边加上

2

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

化简

sec (x) = 2

sec (x) = 2

sec (x) = 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

csc (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

csc (x) + 1 = 0

将

1

到右边

csc (x) + 1 = 0

两边减去

1

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

化简

csc (x) = - 1

csc (x) = - 1

csc (x) = - 1

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{2} + 2 πn

因为方程对以下值无定义:

\frac{3 π}{2} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

2sin(x)=-sqrt(2),0<= x<= 2pi

2 sin (x) = - 2, 0 \leq x \leq 2 π

csc (a) = 3

csc (a) = 5

1-1/2 cos(2x-pi/2)=0

1 - \frac{1}{2} cos (2 x - \frac{π}{2}) = 0

sin(pix)=-sin(pi-pix)

sin (π x) = - sin (π - π x)