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受欢迎的三角函数 >

3tan^2(x)-8tan(x)+5=0

解答

3 tan^{2} (x) - 8 tan (x) + 5 = 0

解答

x = 1.03037 \dots + πn, x = \frac{π}{4} + πn

+1

度数

x = 59.03624 \dots^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

3 tan^{2} (x) - 8 tan (x) + 5 = 0

用替代法求解

3 tan^{2} (x) - 8 tan (x) + 5 = 0

令

: tan (x) = u

3 u^{2} - 8 u + 5 = 0

3 u^{2} - 8 u + 5 = 0 : u = \frac{5}{3}, u = 1

3 u^{2} - 8 u + 5 = 0

使用求根公式求解

3 u^{2} - 8 u + 5 = 0

二次方程求根公式:

若

a = 3, b = - 8, c = 5

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 \cdot 5}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 \cdot 5}{2 \cdot 3}

(- 8)^{2} - 4 \cdot 3 \cdot 5 = 2

(- 8)^{2} - 4 \cdot 3 \cdot 5

使用指数法则:

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

若

n

是偶数

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 3 \cdot 5

数字相乘:

4 \cdot 3 \cdot 5 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

数字相减:

64 - 60 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 8 ) \pm 2}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 8 ) + 2}{2 \cdot 3}, u_{2} = \frac{- ( - 8 ) - 2}{2 \cdot 3}

u = \frac{- ( - 8 ) + 2}{2 \cdot 3} : \frac{5}{3}

\frac{- ( - 8 ) + 2}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 + 2}{2 \cdot 3}

数字相加:

8 + 2 = 10

= \frac{10}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{10}{6}

约分:

2

= \frac{5}{3}

u = \frac{- ( - 8 ) - 2}{2 \cdot 3} : 1

\frac{- ( - 8 ) - 2}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 - 2}{2 \cdot 3}

数字相减:

8 - 2 = 6

= \frac{6}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{6}{6}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = \frac{5}{3}, u = 1

u = tan (x)

代回

tan (x) = \frac{5}{3}, tan (x) = 1

tan (x) = \frac{5}{3}, tan (x) = 1

tan (x) = \frac{5}{3} : x = arctan (\frac{5}{3}) + πn

tan (x) = \frac{5}{3}

使用反三角函数性质

tan (x) = \frac{5}{3}

tan (x) = \frac{5}{3}

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{5}{3}) + πn

x = arctan (\frac{5}{3}) + πn

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

tan (x) = 1

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 = \frac{π}{4} + πn

x = \frac{π}{4} + πn

合并所有解

x = arctan (\frac{5}{3}) + πn, x = \frac{π}{4} + πn

以小数形式表示解

x = 1.03037 \dots + πn, x = \frac{π}{4} + πn

作图

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