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受欢迎的三角函数 >

3tan^2(x)+4tan(x)-3=0

解答

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

解答

x = 0.49139 \dots + πn, x = - 1.07939 \dots + πn

+1

度数

x = 28.15496 \dots^{\circ} + 18 0^{\circ} n, x = - 61.84503 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: tan (x) = u

3 u^{2} + 4 u - 3 = 0

3 u^{2} + 4 u - 3 = 0 : u = \frac{- 2 + 13}{3}, u = - \frac{2 + 13}{3}

3 u^{2} + 4 u - 3 = 0

使用求根公式求解

3 u^{2} + 4 u - 3 = 0

二次方程求根公式:

若

a = 3, b = 4, c = - 3

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

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

4^{2} - 4 \cdot 3 (- 3) = 213

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

使用法则

- (- a) = a

= 4^{2} + 4 \cdot 3 \cdot 3

数字相乘:

4 \cdot 3 \cdot 3 = 36

= 4^{2} + 36

4^{2} = 16

= 16 + 36

数字相加:

16 + 36 = 52

= 52

52

质因数分解:

2^{2} \cdot 13

52

52

除以

252 = 26 \cdot 2

= 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

使用根式运算法则:

n ab = n a n b

= 13 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 213

u_{1, 2} = \frac{- 4 \pm 2 13}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- 4 + 2 13}{2 \cdot 3}, u_{2} = \frac{- 4 - 2 13}{2 \cdot 3}

u = \frac{- 4 + 2 13}{2 \cdot 3} : \frac{- 2 + 13}{3}

\frac{- 4 + 2 13}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 4 + 2 13}{6}

分解

- 4 + 213 : 2 (- 2 + 13)

- 4 + 213

改写为

= - 2 \cdot 2 + 213

因式分解出通项

2

= 2 (- 2 + 13)

= \frac{2 ( - 2 + 13 )}{6}

约分:

2

= \frac{- 2 + 13}{3}

u = \frac{- 4 - 2 13}{2 \cdot 3} : - \frac{2 + 13}{3}

\frac{- 4 - 2 13}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 4 - 2 13}{6}

分解

- 4 - 213 : - 2 (2 + 13)

- 4 - 213

改写为

= - 2 \cdot 2 - 213

因式分解出通项

2

= - 2 (2 + 13)

= - \frac{2 ( 2 + 13 )}{6}

约分:

2

= - \frac{2 + 13}{3}

二次方程组的解是:

u = \frac{- 2 + 13}{3}, u = - \frac{2 + 13}{3}

u = tan (x)

代回

tan (x) = \frac{- 2 + 13}{3}, tan (x) = - \frac{2 + 13}{3}

tan (x) = \frac{- 2 + 13}{3}, tan (x) = - \frac{2 + 13}{3}

tan (x) = \frac{- 2 + 13}{3} : x = arctan (\frac{- 2 + 13}{3}) + πn

tan (x) = \frac{- 2 + 13}{3}

使用反三角函数性质

tan (x) = \frac{- 2 + 13}{3}

tan (x) = \frac{- 2 + 13}{3}

的通解

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

x = arctan (\frac{- 2 + 13}{3}) + πn

x = arctan (\frac{- 2 + 13}{3}) + πn

tan (x) = - \frac{2 + 13}{3} : x = arctan (- \frac{2 + 13}{3}) + πn

tan (x) = - \frac{2 + 13}{3}

使用反三角函数性质

tan (x) = - \frac{2 + 13}{3}

tan (x) = - \frac{2 + 13}{3}

的通解

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

x = arctan (- \frac{2 + 13}{3}) + πn

x = arctan (- \frac{2 + 13}{3}) + πn

合并所有解

x = arctan (\frac{- 2 + 13}{3}) + πn, x = arctan (- \frac{2 + 13}{3}) + πn

以小数形式表示解

x = 0.49139 \dots + πn, x = - 1.07939 \dots + πn

作图

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