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受欢迎的三角函数 >

6sec^2(θ)tan(θ)=12tan(θ)

解答

6 sec^{2} (θ) tan (θ) = 12 tan (θ)

解答

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

+1

度数

θ = 0^{\circ} + 18 0^{\circ} n, θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 22 5^{\circ} + 36 0^{\circ} n, θ = 4 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

6 sec^{2} (θ) tan (θ) = 12 tan (θ)

两边减去

12 tan (θ)

6 sec^{2} (θ) tan (θ) - 12 tan (θ) = 0

分解

6 sec^{2} (θ) tan (θ) - 12 tan (θ) : 6 tan (θ) (sec (θ) + 2) (sec (θ) - 2)

6 sec^{2} (θ) tan (θ) - 12 tan (θ)

将

- 12

改写为

2 \cdot 6

= 6 sec^{2} (θ) tan (θ) + 2 \cdot 6 tan (θ)

因式分解出通项

6 tan (θ)

= 6 tan (θ) (sec^{2} (θ) - 2)

分解

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

sec^{2} (θ) - 2

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

= sec^{2} (θ) - (2)^{2}

使用平方差公式:

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

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

= (sec (θ) + 2) (sec (θ) - 2)

= 6 tan (θ) (sec (θ) + 2) (sec (θ) - 2)

6 tan (θ) (sec (θ) + 2) (sec (θ) - 2) = 0

分别求解每个部分

tan (θ) = 0 or sec (θ) + 2 = 0 or sec (θ) - 2 = 0

tan (θ) = 0 : θ = πn

tan (θ) = 0

tan (θ) = 0

的通解

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = 0 + πn

θ = 0 + πn

解

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

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

sec (θ) + 2 = 0

将

2

到右边

sec (θ) + 2 = 0

两边减去

2

sec (θ) + 2 - 2 = 0 - 2

化简

sec (θ) = - 2

sec (θ) = - 2

sec (θ) = - 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}

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

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

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

sec (θ) - 2 = 0

将

2

到右边

sec (θ) - 2 = 0

两边加上

2

sec (θ) - 2 + 2 = 0 + 2

化简

sec (θ) = 2

sec (θ) = 2

sec (θ) = 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}

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

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

合并所有解

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

作图

流行的例子

7tan(x)=3sqrt(3)+tan(x)

7 tan (x) = 33 + tan (x)

cos(2x)=tan(2x)

cos (2 x) = tan (2 x)

sin (\frac{θ}{2}) - 1 = 0

2sin(4x)+sqrt(3)=0

2 sin (4 x) + 3 = 0

7sec(x)+7tan(x)=7

7 sec (x) + 7 tan (x) = 7