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受欢迎的三角函数 >

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

解答

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

解答

x = 1.24904 \dots + πn, x = \frac{π}{4} + πn

+1

度数

x = 71.56505 \dots^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: tan (x) = u

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

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = - 4, c = 3

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

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

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

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

使用指数法则:

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

若

n

是偶数

(- 4)^{2} = 4^{2}

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

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 4^{2} - 12

4^{2} = 16

= 16 - 12

数字相减:

16 - 12 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

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

将解分隔开

u_{1} = \frac{- ( - 4 ) + 2}{2 \cdot 1}, u_{2} = \frac{- ( - 4 ) - 2}{2 \cdot 1}

u = \frac{- ( - 4 ) + 2}{2 \cdot 1} : 3

\frac{- ( - 4 ) + 2}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{4 + 2}{2 \cdot 1}

数字相加:

4 + 2 = 6

= \frac{6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{6}{2}

数字相除:

\frac{6}{2} = 3

= 3

u = \frac{- ( - 4 ) - 2}{2 \cdot 1} : 1

\frac{- ( - 4 ) - 2}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{4 - 2}{2 \cdot 1}

数字相减:

4 - 2 = 2

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

数字相乘:

2 \cdot 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = 3, u = 1

u = tan (x)

代回

tan (x) = 3, tan (x) = 1

tan (x) = 3, tan (x) = 1

tan (x) = 3 : x = arctan (3) + πn

tan (x) = 3

使用反三角函数性质

tan (x) = 3

tan (x) = 3

的通解

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

x = arctan (3) + πn

x = arctan (3) + πn

tan (x) = 1 : x = \frac{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{π}{4} + πn

x = \frac{π}{4} + πn

合并所有解

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

以小数形式表示解

x = 1.24904 \dots + πn, x = \frac{π}{4} + πn

作图

流行的例子

sin(x)+sin(3x)=cos(x)

sin (x) + sin (3 x) = cos (x)

2sin^2(x)=3(1-cos(x))

2 sin^{2} (x) = 3 (1 - cos (x))

cos (x) = 0, 72

csc (θ) = 2.8

cos(x)=-2/(sqrt(5))

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