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受欢迎的三角函数 >

2cot^2(x)-5=sec^2(x)-tan^2(x)

解答

2 cot^{2} (x) - 5 = sec^{2} (x) - tan^{2} (x)

解答

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

+1

度数

x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

求解步骤

2 cot^{2} (x) - 5 = sec^{2} (x) - tan^{2} (x)

两边减去

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

2 cot^{2} (x) - 5 - sec^{2} (x) + tan^{2} (x) = 0

使用三角恒等式改写

- 5 - sec^{2} (x) + tan^{2} (x) + 2 cot^{2} (x)

使用毕达哥拉斯恒等式:

sec^{2} (x) = tan^{2} (x) + 1

sec^{2} (x) - tan^{2} (x) = 1

= - 5 + 2 cot^{2} (x) - 1

化简

- 5 + 2 cot^{2} (x) - 1 : 2 cot^{2} (x) - 6

- 5 + 2 cot^{2} (x) - 1

对同类项分组

= 2 cot^{2} (x) - 5 - 1

数字相减:

- 5 - 1 = - 6

= 2 cot^{2} (x) - 6

= 2 cot^{2} (x) - 6

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

用替代法求解

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

令

: cot (x) = u

- 6 + 2 u^{2} = 0

- 6 + 2 u^{2} = 0 : u = 3, u = - 3

- 6 + 2 u^{2} = 0

将

6

到右边

- 6 + 2 u^{2} = 0

两边加上

6

- 6 + 2 u^{2} + 6 = 0 + 6

化简

2 u^{2} = 6

2 u^{2} = 6

两边除以

2

2 u^{2} = 6

两边除以

2

\frac{2 u ^{2}}{2} = \frac{6}{2}

化简

u^{2} = 3

u^{2} = 3

对于

x^{2} = f (a)

解为

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

u = 3, u = - 3

u = cot (x)

代回

cot (x) = 3, cot (x) = - 3

cot (x) = 3, cot (x) = - 3

cot (x) = 3 : x = \frac{π}{6} + πn

cot (x) = 3

cot (x) = 3

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}{6} + πn

x = \frac{π}{6} + πn

cot (x) = - 3 : x = \frac{5 π}{6} + πn

cot (x) = - 3

cot (x) = - 3

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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

合并所有解

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

作图

流行的例子

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

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

csc^2(x)-csc(x)-20=0

csc^{2} (x) - csc (x) - 20 = 0

cos(a)=(sqrt(3))/2

cos (a) = \frac{3}{2}

cos (4 t) = 0

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