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受欢迎的三角函数 >

2tan^2(θ)=sec^2(θ)+2

解答

2 tan^{2} (θ) = sec^{2} (θ) + 2

解答

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

+1

度数

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n, θ = 12 0^{\circ} + 36 0^{\circ} n, θ = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

2 tan^{2} (θ) = sec^{2} (θ) + 2

两边减去

sec^{2} (θ) + 2

2 tan^{2} (θ) - sec^{2} (θ) - 2 = 0

使用三角恒等式改写

- 2 - sec^{2} (θ) + 2 tan^{2} (θ)

使用毕达哥拉斯恒等式:

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

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

= - 2 - sec^{2} (θ) + 2 (sec^{2} (θ) - 1)

化简

- 2 - sec^{2} (θ) + 2 (sec^{2} (θ) - 1) : sec^{2} (θ) - 4

- 2 - sec^{2} (θ) + 2 (sec^{2} (θ) - 1)

乘开

2 (sec^{2} (θ) - 1) : 2 sec^{2} (θ) - 2

2 (sec^{2} (θ) - 1)

使用分配律:

a (b - c) = ab - a c

a = 2, b = sec^{2} (θ), c = 1

= 2 sec^{2} (θ) - 2 \cdot 1

数字相乘:

2 \cdot 1 = 2

= 2 sec^{2} (θ) - 2

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

化简

- 2 - sec^{2} (θ) + 2 sec^{2} (θ) - 2 : sec^{2} (θ) - 4

- 2 - sec^{2} (θ) + 2 sec^{2} (θ) - 2

同类项相加:

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

= - 2 + sec^{2} (θ) - 2

对同类项分组

= sec^{2} (θ) - 2 - 2

数字相减:

- 2 - 2 = - 4

= sec^{2} (θ) - 4

= sec^{2} (θ) - 4

= sec^{2} (θ) - 4

- 4 + sec^{2} (θ) = 0

用替代法求解

- 4 + sec^{2} (θ) = 0

令

: sec (θ) = u

- 4 + u^{2} = 0

- 4 + u^{2} = 0 : u = 2, u = - 2

- 4 + u^{2} = 0

将

4

到右边

- 4 + u^{2} = 0

两边加上

4

- 4 + u^{2} + 4 = 0 + 4

化简

u^{2} = 4

u^{2} = 4

对于

x^{2} = f (a)

解为

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

u = 4, u = - 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

u = sec (θ)

代回

sec (θ) = 2, sec (θ) = - 2

sec (θ) = 2, sec (θ) = - 2

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

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}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

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

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

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

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

合并所有解

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

作图

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