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受欢迎的三角函数 >

tan^2(x)+5tan(x)-1=0

解答

tan^{2} (x) + 5 tan (x) - 1 = 0

解答

x = 0.19025 \dots + πn, x = - 1.38054 \dots + πn

+1

度数

x = 10.90070 \dots^{\circ} + 18 0^{\circ} n, x = - 79.09929 \dots^{\circ} + 18 0^{\circ} n

求解步骤

tan^{2} (x) + 5 tan (x) - 1 = 0

用替代法求解

tan^{2} (x) + 5 tan (x) - 1 = 0

令

: tan (x) = u

u^{2} + 5 u - 1 = 0

u^{2} + 5 u - 1 = 0 : u = \frac{- 5 + 29}{2}, u = \frac{- 5 - 29}{2}

u^{2} + 5 u - 1 = 0

使用求根公式求解

u^{2} + 5 u - 1 = 0

二次方程求根公式:

若

a = 1, b = 5, c = - 1

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

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

5^{2} - 4 \cdot 1 \cdot (- 1) = 29

5^{2} - 4 \cdot 1 \cdot (- 1)

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 5^{2} + 4

5^{2} = 25

= 25 + 4

数字相加:

25 + 4 = 29

= 29

u_{1, 2} = \frac{- 5 \pm 29}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 5 + 29}{2 \cdot 1}, u_{2} = \frac{- 5 - 29}{2 \cdot 1}

u = \frac{- 5 + 29}{2 \cdot 1} : \frac{- 5 + 29}{2}

\frac{- 5 + 29}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 5 + 29}{2}

u = \frac{- 5 - 29}{2 \cdot 1} : \frac{- 5 - 29}{2}

\frac{- 5 - 29}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 5 - 29}{2}

二次方程组的解是:

u = \frac{- 5 + 29}{2}, u = \frac{- 5 - 29}{2}

u = tan (x)

代回

tan (x) = \frac{- 5 + 29}{2}, tan (x) = \frac{- 5 - 29}{2}

tan (x) = \frac{- 5 + 29}{2}, tan (x) = \frac{- 5 - 29}{2}

tan (x) = \frac{- 5 + 29}{2} : x = arctan (\frac{- 5 + 29}{2}) + πn

tan (x) = \frac{- 5 + 29}{2}

使用反三角函数性质

tan (x) = \frac{- 5 + 29}{2}

tan (x) = \frac{- 5 + 29}{2}

的通解

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

x = arctan (\frac{- 5 + 29}{2}) + πn

x = arctan (\frac{- 5 + 29}{2}) + πn

tan (x) = \frac{- 5 - 29}{2} : x = arctan (\frac{- 5 - 29}{2}) + πn

tan (x) = \frac{- 5 - 29}{2}

使用反三角函数性质

tan (x) = \frac{- 5 - 29}{2}

tan (x) = \frac{- 5 - 29}{2}

的通解

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

x = arctan (\frac{- 5 - 29}{2}) + πn

x = arctan (\frac{- 5 - 29}{2}) + πn

合并所有解

x = arctan (\frac{- 5 + 29}{2}) + πn, x = arctan (\frac{- 5 - 29}{2}) + πn

以小数形式表示解

x = 0.19025 \dots + πn, x = - 1.38054 \dots + πn

作图

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