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受欢迎的三角函数 >

tan(x)csc(x)-2tan(x)=3csc(x)-6

解答

tan (x) csc (x) - 2 tan (x) = 3 csc (x) - 6

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 1.24904 \dots + πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 71.56505 \dots^{\circ} + 18 0^{\circ} n

求解步骤

tan (x) csc (x) - 2 tan (x) = 3 csc (x) - 6

两边减去

3 csc (x) - 6

tan (x) csc (x) - 2 tan (x) - 3 csc (x) + 6 = 0

分解

tan (x) csc (x) - 2 tan (x) - 3 csc (x) + 6 : (csc (x) - 2) (tan (x) - 3)

tan (x) csc (x) - 2 tan (x) - 3 csc (x) + 6

= (tan (x) csc (x) - 2 tan (x)) + (- 3 csc (x) + 6)

从

- 3 csc (x) + 6

分解出因式

- 3

:

- 3 (csc (x) - 2)

- 3 csc (x) + 6

将

6

改写为

2 \cdot 3

= - 3 csc (x) + 2 \cdot 3

因式分解出通项

- 3

= - 3 (csc (x) - 2)

从

tan (x) csc (x) - 2 tan (x)

分解出因式

tan (x)

:

tan (x) (csc (x) - 2)

tan (x) csc (x) - 2 tan (x)

因式分解出通项

tan (x)

= tan (x) (csc (x) - 2)

= - 3 (csc (x) - 2) + tan (x) (csc (x) - 2)

因式分解出通项

csc (x) - 2

= (csc (x) - 2) (tan (x) - 3)

(csc (x) - 2) (tan (x) - 3) = 0

分别求解每个部分

csc (x) - 2 = 0 or tan (x) - 3 = 0

csc (x) - 2 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) - 2 = 0

将

2

到右边

csc (x) - 2 = 0

两边加上

2

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

化简

csc (x) = 2

csc (x) = 2

csc (x) = 2

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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

tan (x) - 3 = 0 : x = arctan (3) + πn

tan (x) - 3 = 0

将

3

到右边

tan (x) - 3 = 0

两边加上

3

tan (x) - 3 + 3 = 0 + 3

化简

tan (x) = 3

tan (x) = 3

使用反三角函数性质

tan (x) = 3

tan (x) = 3

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (3) + πn

x = arctan (3) + πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = arctan (3) + πn

以小数形式表示解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 1.24904 \dots + πn

作图

流行的例子

10=7cos((pi/6)(t+6))+8

10 = 7 cos ((\frac{π}{6}) (t + 6)) + 8

cos(θ)= 1/(sqrt(6))

cos (θ) = \frac{1}{6}

4tan^2(θ)=-sqrt(3)tan(θ)+tan^2(θ)

4 tan^{2} (θ) = - 3 tan (θ) + tan^{2} (θ)

2sec^2(x)-3tan(x)=11

2 sec^{2} (x) - 3 tan (x) = 11

3 cos (2 x) = - 3