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受欢迎的三角函数 >

tan(x)=-cot(x)

解答

tan (x) = - cot (x)

解答

x \in R 无解

求解步骤

tan (x) = - cot (x)

两边减去

- cot (x)

tan (x) + cot (x) = 0

使用三角恒等式改写

cot (x) + tan (x)

使用基本三角恒等式:

tan (x) = \frac{1}{cot ( x )}

= cot (x) + \frac{1}{cot ( x )}

cot (x) + \frac{1}{cot ( x )} = 0

用替代法求解

cot (x) + \frac{1}{cot ( x )} = 0

令

: cot (x) = u

u + \frac{1}{u} = 0

u + \frac{1}{u} = 0 : u = i, u = - i

u + \frac{1}{u} = 0

在两边乘以

u

u + \frac{1}{u} = 0

在两边乘以

u

uu + \frac{1}{u} u = 0 \cdot u

化简

uu + \frac{1}{u} u = 0 \cdot u

化简

uu : u^{2}

uu

使用指数法则:

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

uu = u^{1 + 1}

= u^{1 + 1}

数字相加:

1 + 1 = 2

= u^{2}

化简

\frac{1}{u} u : 1

\frac{1}{u} u

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot u}{u}

约分:

u

= 1

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

u^{2} + 1 = 0

u^{2} + 1 = 0

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

u = i, u = - i

u = cot (x)

代回

cot (x) = i, cot (x) = - i

cot (x) = i, cot (x) = - i

cot (x) = i :

无解

cot (x) = i

无解

cot (x) = - i :

无解

cot (x) = - i

无解

合并所有解

x \in R 无解

作图

流行的例子

3cos^2(x)-sin^2(x)=0

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

cos (x) = - 0.6

cos (x) = - 0.9

2cos(x)+3sin(x)=0

2 cos (x) + 3 sin (x) = 0

sec(x/5)=sqrt(10)

sec (\frac{x}{5}) = 10