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受欢迎的三角函数 >

3cot(x)=tan(x)

解答

3 cot (x) = tan (x)

解答

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

+1

度数

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n

求解步骤

3 cot (x) = tan (x)

两边减去

tan (x)

3 cot (x) - tan (x) = 0

使用三角恒等式改写

- tan (x) + 3 cot (x)

使用基本三角恒等式:

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

= - \frac{1}{cot ( x )} + 3 cot (x)

- \frac{1}{cot ( x )} + 3 cot (x) = 0

用替代法求解

- \frac{1}{cot ( x )} + 3 cot (x) = 0

令

: cot (x) = u

- \frac{1}{u} + 3 u = 0

- \frac{1}{u} + 3 u = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

- \frac{1}{u} + 3 u = 0

在两边乘以

u

- \frac{1}{u} + 3 u = 0

在两边乘以

u

- \frac{1}{u} u + 3 uu = 0 \cdot u

化简

- \frac{1}{u} u + 3 uu = 0 \cdot u

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

3 uu : 3 u^{2}

3 uu

使用指数法则:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

数字相加:

1 + 1 = 2

= 3 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- 1 + 3 u^{2} = 0

- 1 + 3 u^{2} = 0

- 1 + 3 u^{2} = 0

解

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

- 1 + 3 u^{2} = 0

将

1

到右边

- 1 + 3 u^{2} = 0

两边加上

1

- 1 + 3 u^{2} + 1 = 0 + 1

化简

3 u^{2} = 1

3 u^{2} = 1

两边除以

3

3 u^{2} = 1

两边除以

3

\frac{3 u ^{2}}{3} = \frac{1}{3}

化简

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

对于

x^{2} = f (a)

解为

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

u = \frac{1}{3}, u = - \frac{1}{3}

u = \frac{1}{3}, u = - \frac{1}{3}

验证解

找到无定义的点（奇点）:

u = 0

取

- \frac{1}{u} + 3 u

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{1}{3}, u = - \frac{1}{3}

u = cot (x)

代回

cot (x) = \frac{1}{3}, cot (x) = - \frac{1}{3}

cot (x) = \frac{1}{3}, cot (x) = - \frac{1}{3}

cot (x) = \frac{1}{3} : x = arccot (\frac{1}{3}) + πn

cot (x) = \frac{1}{3}

使用反三角函数性质

cot (x) = \frac{1}{3}

cot (x) = \frac{1}{3}

的通解

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (\frac{1}{3}) + πn

x = arccot (\frac{1}{3}) + πn

cot (x) = - \frac{1}{3} : x = arccot (- \frac{1}{3}) + πn

cot (x) = - \frac{1}{3}

使用反三角函数性质

cot (x) = - \frac{1}{3}

cot (x) = - \frac{1}{3}

的通解

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- \frac{1}{3}) + πn

x = arccot (- \frac{1}{3}) + πn

合并所有解

x = arccot (\frac{1}{3}) + πn, x = arccot (- \frac{1}{3}) + πn

以小数形式表示解

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

作图

流行的例子

cot(θ)+2csc(θ)=3

cot (θ) + 2 csc (θ) = 3

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

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

2sin^2(x)-4sin(x)-6=0

2 sin^{2} (x) - 4 sin (x) - 6 = 0

cos^2(2x)= 1/2-1/2 cos(2x)

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

2sin(A)-sqrt(3)=0

2 sin (A) - 3 = 0