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受欢迎的三角函数 >

3tan^2(x)+7tan(x)+4=0

解答

3 tan^{2} (x) + 7 tan (x) + 4 = 0

解答

x = \frac{3 π}{4} + πn, x = - 0.92729 \dots + πn

+1

度数

x = 13 5^{\circ} + 18 0^{\circ} n, x = - 53.13010 \dots^{\circ} + 18 0^{\circ} n

求解步骤

3 tan^{2} (x) + 7 tan (x) + 4 = 0

用替代法求解

3 tan^{2} (x) + 7 tan (x) + 4 = 0

令

: tan (x) = u

3 u^{2} + 7 u + 4 = 0

3 u^{2} + 7 u + 4 = 0 : u = - 1, u = - \frac{4}{3}

3 u^{2} + 7 u + 4 = 0

使用求根公式求解

3 u^{2} + 7 u + 4 = 0

二次方程求根公式:

若

a = 3, b = 7, c = 4

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

7^{2} - 4 \cdot 3 \cdot 4 = 1

7^{2} - 4 \cdot 3 \cdot 4

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

数字相减:

49 - 48 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 7 \pm 1}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- 7 + 1}{2 \cdot 3}, u_{2} = \frac{- 7 - 1}{2 \cdot 3}

u = \frac{- 7 + 1}{2 \cdot 3} : - 1

\frac{- 7 + 1}{2 \cdot 3}

数字相加/相减:

- 7 + 1 = - 6

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

数字相乘:

2 \cdot 3 = 6

= \frac{- 6}{6}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{6}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- 7 - 1}{2 \cdot 3} : - \frac{4}{3}

\frac{- 7 - 1}{2 \cdot 3}

数字相减:

- 7 - 1 = - 8

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

数字相乘:

2 \cdot 3 = 6

= \frac{- 8}{6}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{8}{6}

约分:

2

= - \frac{4}{3}

二次方程组的解是:

u = - 1, u = - \frac{4}{3}

u = tan (x)

代回

tan (x) = - 1, tan (x) = - \frac{4}{3}

tan (x) = - 1, tan (x) = - \frac{4}{3}

tan (x) = - 1 : x = \frac{3 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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

tan (x) = - \frac{4}{3} : x = arctan (- \frac{4}{3}) + πn

tan (x) = - \frac{4}{3}

使用反三角函数性质

tan (x) = - \frac{4}{3}

tan (x) = - \frac{4}{3}

的通解

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

x = arctan (- \frac{4}{3}) + πn

x = arctan (- \frac{4}{3}) + πn

合并所有解

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

以小数形式表示解

x = \frac{3 π}{4} + πn, x = - 0.92729 \dots + πn

作图

流行的例子

cos(2θ)-3=-5sin(θ)

cos (2 θ) - 3 = - 5 sin (θ)

sin(sqrt(2)x)=0

sin (2 x) = 0

7sin^2(x)+3cos^2(x)=5

7 sin^{2} (x) + 3 cos^{2} (x) = 5

sin (x) = 0.3746

tan (a) = 0.49