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受欢迎的三角函数 >

tan(θ)-3cot(θ)=0

解答

tan (θ) - 3 cot (θ) = 0

解答

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

+1

度数

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

求解步骤

tan (θ) - 3 cot (θ) = 0

使用三角恒等式改写

tan (θ) - 3 cot (θ)

使用基本三角恒等式:

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

= \frac{1}{cot ( θ )} - 3 cot (θ)

\frac{1}{cot ( θ )} - 3 cot (θ) = 0

用替代法求解

\frac{1}{cot ( θ )} - 3 cot (θ) = 0

令

: cot (θ) = 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

两边减去

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 (θ)

代回

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

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

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

cot (θ) = \frac{1}{3}

使用反三角函数性质

cot (θ) = \frac{1}{3}

cot (θ) = \frac{1}{3}

的通解

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

θ = arccot (\frac{1}{3}) + πn

θ = arccot (\frac{1}{3}) + πn

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

cot (θ) = - \frac{1}{3}

使用反三角函数性质

cot (θ) = - \frac{1}{3}

cot (θ) = - \frac{1}{3}

的通解

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

θ = arccot (- \frac{1}{3}) + πn

θ = arccot (- \frac{1}{3}) + πn

合并所有解

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

以小数形式表示解

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

作图

流行的例子

cot (- \frac{π}{2})

sin^2(x)+sin(x)=0

sin^{2} (x) + sin (x) = 0

sin (1 5^{\circ})

sec (12 0^{\circ})

csc (9 0^{\circ})