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受欢迎的三角函数 >

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

解答

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

解答

x = 0.87300 \dots + πn, x = - 1.33665 \dots + πn

+1

度数

x = 50.01961 \dots^{\circ} + 18 0^{\circ} n, x = - 76.58466 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: tan (x) = u

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

u^{2} + 3 u - 5 = 0 : u = \frac{- 3 + 29}{2}, u = \frac{- 3 - 29}{2}

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = 3, c = - 5

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

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

3^{2} - 4 \cdot 1 \cdot (- 5) = 29

3^{2} - 4 \cdot 1 \cdot (- 5)

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 1 \cdot 5

数字相乘:

4 \cdot 1 \cdot 5 = 20

= 3^{2} + 20

3^{2} = 9

= 9 + 20

数字相加:

9 + 20 = 29

= 29

u_{1, 2} = \frac{- 3 \pm 29}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 3 + 29}{2 \cdot 1}, u_{2} = \frac{- 3 - 29}{2 \cdot 1}

u = \frac{- 3 + 29}{2 \cdot 1} : \frac{- 3 + 29}{2}

\frac{- 3 + 29}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 3 + 29}{2}

u = \frac{- 3 - 29}{2 \cdot 1} : \frac{- 3 - 29}{2}

\frac{- 3 - 29}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 3 - 29}{2}

二次方程组的解是:

u = \frac{- 3 + 29}{2}, u = \frac{- 3 - 29}{2}

u = tan (x)

代回

tan (x) = \frac{- 3 + 29}{2}, tan (x) = \frac{- 3 - 29}{2}

tan (x) = \frac{- 3 + 29}{2}, tan (x) = \frac{- 3 - 29}{2}

tan (x) = \frac{- 3 + 29}{2} : x = arctan (\frac{- 3 + 29}{2}) + πn

tan (x) = \frac{- 3 + 29}{2}

使用反三角函数性质

tan (x) = \frac{- 3 + 29}{2}

tan (x) = \frac{- 3 + 29}{2}

的通解

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

x = arctan (\frac{- 3 + 29}{2}) + πn

x = arctan (\frac{- 3 + 29}{2}) + πn

tan (x) = \frac{- 3 - 29}{2} : x = arctan (\frac{- 3 - 29}{2}) + πn

tan (x) = \frac{- 3 - 29}{2}

使用反三角函数性质

tan (x) = \frac{- 3 - 29}{2}

tan (x) = \frac{- 3 - 29}{2}

的通解

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

x = arctan (\frac{- 3 - 29}{2}) + πn

x = arctan (\frac{- 3 - 29}{2}) + πn

合并所有解

x = arctan (\frac{- 3 + 29}{2}) + πn, x = arctan (\frac{- 3 - 29}{2}) + πn

以小数形式表示解

x = 0.87300 \dots + πn, x = - 1.33665 \dots + πn

作图

流行的例子

tan(1/2 x)-sqrt(3)=0

tan (\frac{1}{2} x) - 3 = 0

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2 cos (2 θ) = 6 cos (θ) - 4

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cos(2θ)-cos(4θ)=0

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