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受欢迎的三角函数 >

3tan^2(x)+5=7sec(x)

解答

3 tan^{2} (x) + 5 = 7 sec (x)

解答

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

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

3 tan^{2} (x) + 5 = 7 sec (x)

两边减去

7 sec (x)

3 tan^{2} (x) + 5 - 7 sec (x) = 0

使用三角恒等式改写

5 + 3 tan^{2} (x) - 7 sec (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

= 5 + 3 (sec^{2} (x) - 1) - 7 sec (x)

化简

5 + 3 (sec^{2} (x) - 1) - 7 sec (x) : 3 sec^{2} (x) - 7 sec (x) + 2

5 + 3 (sec^{2} (x) - 1) - 7 sec (x)

乘开

3 (sec^{2} (x) - 1) : 3 sec^{2} (x) - 3

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

使用分配律:

a (b - c) = ab - a c

a = 3, b = sec^{2} (x), c = 1

= 3 sec^{2} (x) - 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

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

= 5 + 3 sec^{2} (x) - 3 - 7 sec (x)

化简

5 + 3 sec^{2} (x) - 3 - 7 sec (x) : 3 sec^{2} (x) - 7 sec (x) + 2

5 + 3 sec^{2} (x) - 3 - 7 sec (x)

对同类项分组

= 3 sec^{2} (x) - 7 sec (x) + 5 - 3

数字相加/相减:

5 - 3 = 2

= 3 sec^{2} (x) - 7 sec (x) + 2

= 3 sec^{2} (x) - 7 sec (x) + 2

= 3 sec^{2} (x) - 7 sec (x) + 2

2 + 3 sec^{2} (x) - 7 sec (x) = 0

用替代法求解

2 + 3 sec^{2} (x) - 7 sec (x) = 0

令

: sec (x) = u

2 + 3 u^{2} - 7 u = 0

2 + 3 u^{2} - 7 u = 0 : u = 2, u = \frac{1}{3}

2 + 3 u^{2} - 7 u = 0

改写成标准形式

a x^{2} + b x + c = 0

3 u^{2} - 7 u + 2 = 0

使用求根公式求解

3 u^{2} - 7 u + 2 = 0

二次方程求根公式:

若

a = 3, b = - 7, c = 2

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 2}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 2}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 \cdot 2 = 5

(- 7)^{2} - 4 \cdot 3 \cdot 2

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 2

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

数字相减:

49 - 24 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 5}{2 \cdot 3}, u_{2} = \frac{- ( - 7 ) - 5}{2 \cdot 3}

u = \frac{- ( - 7 ) + 5}{2 \cdot 3} : 2

\frac{- ( - 7 ) + 5}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 + 5}{2 \cdot 3}

数字相加:

7 + 5 = 12

= \frac{12}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{12}{6}

数字相除:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 7 ) - 5}{2 \cdot 3} : \frac{1}{3}

\frac{- ( - 7 ) - 5}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 - 5}{2 \cdot 3}

数字相减:

7 - 5 = 2

= \frac{2}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{2}{6}

约分:

2

= \frac{1}{3}

二次方程组的解是:

u = 2, u = \frac{1}{3}

u = sec (x)

代回

sec (x) = 2, sec (x) = \frac{1}{3}

sec (x) = 2, sec (x) = \frac{1}{3}

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

sec (x) = 2

sec (x) = 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}

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

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

sec (x) = \frac{1}{3} :

无解

sec (x) = \frac{1}{3}

sec (x) \leq - 1 or sec (x) \geq 1

无解

合并所有解

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

作图

流行的例子

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