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受欢迎的三角函数 >

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

解答

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

解答

x = πn

+1

度数

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

求解步骤

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

用替代法求解

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

令

: tan (x) = u

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

u^{5} + u^{3} = 0 : u = 0, u = i, u = - i

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

因式分解

u^{5} + u^{3} : u^{3} (u^{2} + 1)

u^{5} + u^{3}

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{5} = u^{2} u^{3}

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

因式分解出通项

u^{3}

= u^{3} (u^{2} + 1)

u^{3} (u^{2} + 1) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or u^{2} + 1 = 0

解

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

u^{2} + 1 = 0

将

1

到右边

u^{2} + 1 = 0

两边减去

1

u^{2} + 1 - 1 = 0 - 1

化简

u^{2} = - 1

u^{2} = - 1

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = - 1, u = - - 1

化简

- 1 : i

- 1

使用虚数运算法则:

- 1 = i

= i

化简

- - 1 : - i

- - 1

使用虚数运算法则:

- 1 = i

= - i

u = i, u = - i

解为

u = 0, u = i, u = - i

u = tan (x)

代回

tan (x) = 0, tan (x) = i, tan (x) = - i

tan (x) = 0, tan (x) = i, tan (x) = - i

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

tan (x) = i :

无解

tan (x) = i

无解

tan (x) = - i :

无解

tan (x) = - i

无解

合并所有解

x = πn

作图

流行的例子

0=cos(x)(1-cos(x))+sin^2(x)

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

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

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

1/(cos(x))=1.98

\frac{1}{cos ( x )} = 1.98

2sin^2(3θ)-1=0

2 sin^{2} (3 θ) - 1 = 0

4sin(4θ-pi/3)+3=5

4 sin (4 θ - \frac{π}{3}) + 3 = 5