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受欢迎的三角函数 >

tan(x)sin^2(x)=3tan(x)

解答

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

解答

x = πn

+1

度数

x = 0^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

3 tan (x)

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

分解

sin^{2} (x) tan (x) - 3 tan (x) : tan (x) (sin (x) + 3) (sin (x) - 3)

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

因式分解出通项

tan (x)

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

分解

sin^{2} (x) - 3 : (sin (x) + 3) (sin (x) - 3)

sin^{2} (x) - 3

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - (3)^{2} = (sin (x) + 3) (sin (x) - 3)

= (sin (x) + 3) (sin (x) - 3)

= tan (x) (sin (x) + 3) (sin (x) - 3)

tan (x) (sin (x) + 3) (sin (x) - 3) = 0

分别求解每个部分

tan (x) = 0 or sin (x) + 3 = 0 or sin (x) - 3 = 0

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 0

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 = 0 + πn

x = 0 + πn

解

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sin (x) + 3 = 0 :

无解

sin (x) + 3 = 0

将

3

到右边

sin (x) + 3 = 0

两边减去

3

sin (x) + 3 - 3 = 0 - 3

化简

sin (x) = - 3

sin (x) = - 3

- 1 \leq sin (x) \leq 1

无解

sin (x) - 3 = 0 :

无解

sin (x) - 3 = 0

将

3

到右边

sin (x) - 3 = 0

两边加上

3

sin (x) - 3 + 3 = 0 + 3

化简

sin (x) = 3

sin (x) = 3

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = πn

作图

流行的例子

tan^2(x)-sqrt(3)tan(x)=0

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

sin (x) = tan (x)

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

cos (\frac{x}{2}) = \frac{1}{2}

csc(x)+cot(x)=1,0<= x<= 2pi

csc (x) + cot (x) = 1, 0 \leq x \leq 2 π