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受欢迎的三角函数 >

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

解答

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

解答

x = 1.30113 \dots + πn, x = 0.94440 \dots + πn

+1

度数

x = 74.54956 \dots^{\circ} + 18 0^{\circ} n, x = 54.11024 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: tan (x) = u

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

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

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

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

使用指数法则:

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

若

n

是偶数

(- 5)^{2} = 5^{2}

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

数字相乘:

4 \cdot 1 \cdot 5 = 20

= 5^{2} - 20

5^{2} = 25

= 25 - 20

数字相减:

25 - 20 = 5

= 5

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

将解分隔开

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

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

\frac{- ( - 5 ) + 5}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{5 + 5}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{5 + 5}{2}

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

\frac{- ( - 5 ) - 5}{2 \cdot 1}

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

= \frac{5 - 5}{2}

二次方程组的解是:

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

u = tan (x)

代回

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

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

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (\frac{5 + 5}{2}) + πn

x = arctan (\frac{5 + 5}{2}) + πn

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (\frac{5 - 5}{2}) + πn

x = arctan (\frac{5 - 5}{2}) + πn

合并所有解

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

以小数形式表示解

x = 1.30113 \dots + πn, x = 0.94440 \dots + πn

作图

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