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受欢迎的三角函数 >

3sec^2(x)tan(x)=4tan(x)

解答

3 sec^{2} (x) tan (x) = 4 tan (x)

解答

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

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

3 sec^{2} (x) tan (x) = 4 tan (x)

两边减去

4 tan (x)

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

分解

3 sec^{2} (x) tan (x) - 4 tan (x) : tan (x) (3 sec (x) + 2) (3 sec (x) - 2)

3 sec^{2} (x) tan (x) - 4 tan (x)

因式分解出通项

tan (x)

= tan (x) (3 sec^{2} (x) - 4)

分解

3 sec^{2} (x) - 4 : (3 sec (x) + 2) (3 sec (x) - 2)

3 sec^{2} (x) - 4

将

3 sec^{2} (x) - 4

改写为

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

3 sec^{2} (x) - 4

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} sec^{2} (x) - 4

将

4

改写为

2^{2}

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(3)^{2} sec^{2} (x) = (3 sec (x))^{2}

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

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

使用平方差公式:

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

(3 sec (x))^{2} - 2^{2} = (3 sec (x) + 2) (3 sec (x) - 2)

= (3 sec (x) + 2) (3 sec (x) - 2)

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

tan (x) (3 sec (x) + 2) (3 sec (x) - 2) = 0

分别求解每个部分

tan (x) = 0 or 3 sec (x) + 2 = 0 or 3 sec (x) - 2 = 0

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 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}

x = 0 + πn

x = 0 + πn

解

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

3 sec (x) + 2 = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

3 sec (x) + 2 = 0

将

2

到右边

3 sec (x) + 2 = 0

两边减去

2

3 sec (x) + 2 - 2 = 0 - 2

化简

3 sec (x) = - 2

3 sec (x) = - 2

两边除以

3

3 sec (x) = - 2

两边除以

3

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

化简

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

化简

\frac{3 sec ( x )}{3} : sec (x)

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

约分:

3

= sec (x)

化简

\frac{- 2}{3} : - \frac{2 3}{3}

\frac{- 2}{3}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \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}

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

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

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

3 sec (x) - 2 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

3 sec (x) - 2 = 0

将

2

到右边

3 sec (x) - 2 = 0

两边加上

2

3 sec (x) - 2 + 2 = 0 + 2

化简

3 sec (x) = 2

3 sec (x) = 2

两边除以

3

3 sec (x) = 2

两边除以

3

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

化简

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

化简

\frac{3 sec ( x )}{3} : sec (x)

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

约分:

3

= sec (x)

化简

\frac{2}{3} : \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}

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

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

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

合并所有解

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

作图

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