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受欢迎的三角函数 >

5tan(x)=15cot(x)

解答

5 tan (x) = 15 cot (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

5 tan (x) = 15 cot (x)

两边减去

15 cot (x)

5 tan (x) - 15 cot (x) = 0

使用三角恒等式改写

- 15 cot (x) + 5 tan (x)

使用基本三角恒等式:

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

= - 15 cot (x) + 5 \cdot \frac{1}{cot ( x )}

5 \cdot \frac{1}{cot ( x )} = \frac{5}{cot ( x )}

5 \cdot \frac{1}{cot ( x )}

分式相乘:

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

= \frac{1 \cdot 5}{cot ( x )}

数字相乘:

1 \cdot 5 = 5

= \frac{5}{cot ( x )}

= - 15 cot (x) + \frac{5}{cot ( x )}

\frac{5}{cot ( x )} - 15 cot (x) = 0

用替代法求解

\frac{5}{cot ( x )} - 15 cot (x) = 0

令

: cot (x) = u

\frac{5}{u} - 15 u = 0

\frac{5}{u} - 15 u = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

\frac{5}{u} - 15 u = 0

在两边乘以

u

\frac{5}{u} - 15 u = 0

在两边乘以

u

\frac{5}{u} u - 15 uu = 0 \cdot u

化简

\frac{5}{u} u - 15 uu = 0 \cdot u

化简

\frac{5}{u} u : 5

\frac{5}{u} u

分式相乘:

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

= \frac{5 u}{u}

约分:

u

= 5

化简

- 15 uu : - 15 u^{2}

- 15 uu

使用指数法则:

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

uu = u^{1 + 1}

= - 15 u^{1 + 1}

数字相加:

1 + 1 = 2

= - 15 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

5 - 15 u^{2} = 0

5 - 15 u^{2} = 0

5 - 15 u^{2} = 0

解

5 - 15 u^{2} = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

5 - 15 u^{2} = 0

将

5

到右边

5 - 15 u^{2} = 0

两边减去

5

5 - 15 u^{2} - 5 = 0 - 5

化简

- 15 u^{2} = - 5

- 15 u^{2} = - 5

两边除以

- 15

- 15 u^{2} = - 5

两边除以

- 15

\frac{- 15 u ^{2}}{- 15} = \frac{- 5}{- 15}

化简

\frac{- 15 u ^{2}}{- 15} = \frac{- 5}{- 15}

化简

\frac{- 15 u ^{2}}{- 15} : u^{2}

\frac{- 15 u ^{2}}{- 15}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{15 u ^{2}}{15}

数字相除:

\frac{15}{15} = 1

= u^{2}

化简

\frac{- 5}{- 15} : \frac{1}{3}

\frac{- 5}{- 15}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{5}{15}

约分:

5

= \frac{1}{3}

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

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{5}{u} - 15 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

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