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受欢迎的三角函数 >

4sec^2(x)+2tan^2(x)-6=0

解答

4 sec^{2} (x) + 2 tan^{2} (x) - 6 = 0

解答

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sec^{2} (x) + 2 tan^{2} (x) - 6 = 0

使用三角恒等式改写

- 6 + 2 tan^{2} (x) + 4 sec^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 6 + 2 (sec^{2} (x) - 1) + 4 sec^{2} (x)

化简

- 6 + 2 (sec^{2} (x) - 1) + 4 sec^{2} (x) : 6 sec^{2} (x) - 8

- 6 + 2 (sec^{2} (x) - 1) + 4 sec^{2} (x)

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = 2, b = sec^{2} (x), c = 1

= 2 sec^{2} (x) - 2 \cdot 1

数字相乘:

2 \cdot 1 = 2

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

= - 6 + 2 sec^{2} (x) - 2 + 4 sec^{2} (x)

化简

- 6 + 2 sec^{2} (x) - 2 + 4 sec^{2} (x) : 6 sec^{2} (x) - 8

- 6 + 2 sec^{2} (x) - 2 + 4 sec^{2} (x)

对同类项分组

= 2 sec^{2} (x) + 4 sec^{2} (x) - 6 - 2

同类项相加:

2 sec^{2} (x) + 4 sec^{2} (x) = 6 sec^{2} (x)

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

数字相减:

- 6 - 2 = - 8

= 6 sec^{2} (x) - 8

= 6 sec^{2} (x) - 8

= 6 sec^{2} (x) - 8

- 8 + 6 sec^{2} (x) = 0

用替代法求解

- 8 + 6 sec^{2} (x) = 0

令

: sec (x) = u

- 8 + 6 u^{2} = 0

- 8 + 6 u^{2} = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

- 8 + 6 u^{2} = 0

将

8

到右边

- 8 + 6 u^{2} = 0

两边加上

8

- 8 + 6 u^{2} + 8 = 0 + 8

化简

6 u^{2} = 8

6 u^{2} = 8

两边除以

6

6 u^{2} = 8

两边除以

6

\frac{6 u ^{2}}{6} = \frac{8}{6}

化简

u^{2} = \frac{4}{3}

u^{2} = \frac{4}{3}

对于

x^{2} = f (a)

解为

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

u = \frac{4}{3}, u = - \frac{4}{3}

\frac{4}{3} = \frac{2 3}{3}

\frac{4}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{4}{3}

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

= \frac{2}{3}

\frac{2}{3}

有理化:

\frac{2 3}{3}

\frac{2}{3}

乘以共轭根式

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

- \frac{4}{3} = - \frac{2 3}{3}

- \frac{4}{3}

化简

\frac{4}{3} : \frac{2}{3}

\frac{4}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{4}{3}

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

= \frac{2}{3}

= - \frac{2}{3}

- \frac{2}{3}

有理化:

- \frac{2 3}{3}

- \frac{2}{3}

乘以共轭根式

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

u = sec (x)

代回

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}

sec (x) = \frac{2 3}{3} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = \frac{2 3}{3}

sec (x) = \frac{2 3}{3}

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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3}

sec (x) = - \frac{2 3}{3}

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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

作图

流行的例子

tan (θ) = - \frac{1}{2}

4 sin^{2} (θ) - 2 = 0

2tan(θ)cos^2(θ)=1

2 tan (θ) cos^{2} (θ) = 1

sin (θ) = - \frac{3}{4}

4cos^2(θ)-4cos(θ)+1=0

4 cos^{2} (θ) - 4 cos (θ) + 1 = 0