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受欢迎的三角函数 >

2tan(x)=3-tan^2(x),-2pi<= ,x<= pi

解答

2 tan (x) = 3 - tan^{2} (x), - 2 π \leq, x \leq π

解答

x \in R 无解

求解步骤

2 tan (x) = 3 - tan^{2} (x), - 2 π, x \leq π

用替代法求解

2 tan (x) = 3 - tan^{2} (x)

令

: tan (x) = u

2 u = 3 - u^{2}

2 u = 3 - u^{2} : u = - 3, u = 1

2 u = 3 - u^{2}

交换两边

3 - u^{2} = 2 u

将

2 u

para o lado esquerdo

3 - u^{2} = 2 u

两边减去

2 u

3 - u^{2} - 2 u = 2 u - 2 u

化简

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

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

改写成标准形式

a x^{2} + b x + c = 0

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 1, b = - 2, c = 3

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

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

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

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

数字相加:

4 + 12 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

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

将解分隔开

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

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

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

去除括号:

(- a) = - a, - (- a) = a

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

数字相加:

2 + 4 = 6

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

数字相乘:

2 \cdot 1 = 2

= \frac{6}{- 2}

使用分式法则:

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

= - \frac{6}{2}

数字相除:

\frac{6}{2} = 3

= - 3

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

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

去除括号:

(- a) = - a, - (- a) = a

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

数字相减:

2 - 4 = - 2

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 2}{- 2}

使用分式法则:

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

= \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, - 2 π :

无解

tan (x) = - 3, - 2 π

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

在

- 2 π

范围内的解

无解

tan (x) = 1, - 2 π :

无解

tan (x) = 1, - 2 π

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

在

- 2 π

范围内的解

无解

合并所有解

x \in R 无解

作图

流行的例子

-25=(29cos(x))*3.7-4.9*3.7^2

- 25 = (29 cos (x)) \cdot 3.7 - 4.9 \cdot 3. 7^{2}

solvefor t,x=5(sin(t)+cos(t))

so l v e f or t, x = 5 (sin (t) + cos (t))

2/2*sqrt(3)*cos(x)-2/2*sin(x)=0

\frac{2}{2} \cdot 3 \cdot cos (x) - \frac{2}{2} \cdot sin (x) = 0

solvefor a,sin(a)=0.5

so l v e f or a, sin (a) = 0.5

3/(2cos(θ))=3-3cos(θ)

\frac{3}{2 cos ( θ )} = 3 - 3 cos (θ)