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受欢迎的三角函数 >

1+cot^2(x)=8sin(x)

解答

1 + cot^{2} (x) = 8 sin (x)

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

1 + cot^{2} (x) = 8 sin (x)

两边减去

8 sin (x)

1 + cot^{2} (x) - 8 sin (x) = 0

使用三角恒等式改写

1 + cot^{2} (x) - 8 sin (x)

使用基本三角恒等式:

cot (x) = \frac{cos ( x )}{sin ( x )}

= 1 + (\frac{cos ( x )}{sin ( x )})^{2} - 8 sin (x)

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= 1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x)

使用毕达哥拉斯恒等式:

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

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

= 1 + \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x)

1 + \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x) = 0

用替代法求解

1 + \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x) = 0

令

: sin (x) = u

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0 : u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0

在两边乘以

u^{2}

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0

在两边乘以

u^{2}

1 \cdot u^{2} + \frac{1 - u ^{2}}{u ^{2}} u^{2} - 8 u u^{2} = 0 \cdot u^{2}

化简

1 \cdot u^{2} + \frac{1 - u ^{2}}{u ^{2}} u^{2} - 8 u u^{2} = 0 \cdot u^{2}

化简

1 \cdot u^{2} : u^{2}

1 \cdot u^{2}

乘以:

1 \cdot u^{2} = u^{2}

= u^{2}

化简

\frac{1 - u ^{2}}{u ^{2}} u^{2} : 1 - u^{2}

\frac{1 - u ^{2}}{u ^{2}} u^{2}

分式相乘:

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

= \frac{( 1 - u ^{2} ) u ^{2}}{u ^{2}}

约分:

u^{2}

= 1 - u^{2}

化简

- 8 u u^{2} : - 8 u^{3}

- 8 u u^{2}

使用指数法则:

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

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

= - 8 u^{1 + 2}

数字相加:

1 + 2 = 3

= - 8 u^{3}

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

u^{2} + 1 - u^{2} - 8 u^{3} = 0

化简

u^{2} + 1 - u^{2} - 8 u^{3} : - 8 u^{3} + 1

u^{2} + 1 - u^{2} - 8 u^{3}

对同类项分组

= - 8 u^{3} + u^{2} - u^{2} + 1

同类项相加:

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

= - 8 u^{3} + 1

- 8 u^{3} + 1 = 0

- 8 u^{3} + 1 = 0

- 8 u^{3} + 1 = 0

解

- 8 u^{3} + 1 = 0 : u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

- 8 u^{3} + 1 = 0

将

1

到右边

- 8 u^{3} + 1 = 0

两边减去

1

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

化简

- 8 u^{3} = - 1

- 8 u^{3} = - 1

两边除以

- 8

- 8 u^{3} = - 1

两边除以

- 8

\frac{- 8 u ^{3}}{- 8} = \frac{- 1}{- 8}

化简

u^{3} = \frac{1}{8}

u^{3} = \frac{1}{8}

对于

x^{3} = f (a)

解为

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 \frac{1}{8}, u = 3 \frac{1}{8} \frac{- 1 + 3 i}{2}, u = 3 \frac{1}{8} \frac{- 1 - 3 i}{2}

3 \frac{1}{8} = \frac{1}{2}

3 \frac{1}{8}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 8}

38 = 2

38

因式分解数字:

8 = 2^{3}

= 3 2^{3}

使用根式运算法则:

n a^{n} = a

3 2^{3} = 2

= 2

= \frac{3 1}{2}

使用法则

n 1 = 1

31 = 1

= \frac{1}{2}

化简

3 \frac{1}{8} \frac{- 1 + 3 i}{2} : - \frac{1}{4} + i \frac{3}{4}

3 \frac{1}{8} \frac{- 1 + 3 i}{2}

3 \frac{1}{8} = \frac{1}{2}

3 \frac{1}{8}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 8}

38 = 2

38

因式分解数字:

8 = 2^{3}

= 3 2^{3}

使用根式运算法则:

n a^{n} = a

3 2^{3} = 2

= 2

= \frac{3 1}{2}

使用法则

n 1 = 1

31 = 1

= \frac{1}{2}

= \frac{1}{2} \cdot \frac{- 1 + 3 i}{2}

分式相乘:

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

= \frac{1 \cdot ( - 1 + 3 i )}{2 \cdot 2}

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

乘以:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

去除括号:

(- a) = - a

= - 1 + 3 i

= \frac{- 1 + 3 i}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 1 + 3 i}{4}

将

\frac{- 1 + 3 i}{4}

改写成标准复数形式:

- \frac{1}{4} + \frac{3}{4} i

\frac{- 1 + 3 i}{4}

使用分式法则:

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

\frac{- 1 + 3 i}{4} = - \frac{1}{4} + \frac{3 i}{4}

= - \frac{1}{4} + \frac{3 i}{4}

= - \frac{1}{4} + \frac{3}{4} i

化简

3 \frac{1}{8} \frac{- 1 - 3 i}{2} : - \frac{1}{4} - i \frac{3}{4}

3 \frac{1}{8} \frac{- 1 - 3 i}{2}

3 \frac{1}{8} = \frac{1}{2}

3 \frac{1}{8}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 8}

38 = 2

38

因式分解数字:

8 = 2^{3}

= 3 2^{3}

使用根式运算法则:

n a^{n} = a

3 2^{3} = 2

= 2

= \frac{3 1}{2}

使用法则

n 1 = 1

31 = 1

= \frac{1}{2}

= \frac{1}{2} \cdot \frac{- 1 - 3 i}{2}

分式相乘:

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

= \frac{1 \cdot ( - 1 - 3 i )}{2 \cdot 2}

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

乘以:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

去除括号:

(- a) = - a

= - 1 - 3 i

= \frac{- 1 - 3 i}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 1 - 3 i}{4}

将

\frac{- 1 - 3 i}{4}

改写成标准复数形式:

- \frac{1}{4} - \frac{3}{4} i

\frac{- 1 - 3 i}{4}

使用分式法则:

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

\frac{- 1 - 3 i}{4} = - \frac{1}{4} - \frac{3 i}{4}

= - \frac{1}{4} - \frac{3 i}{4}

= - \frac{1}{4} - \frac{3}{4} i

u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

验证解

找到无定义的点（奇点）:

u = 0

取

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

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}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

u = sin (x)

代回

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{4} + i \frac{3}{4}, sin (x) = - \frac{1}{4} - i \frac{3}{4}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{4} + i \frac{3}{4}, sin (x) = - \frac{1}{4} - i \frac{3}{4}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

的通解

sin (x)

周期表（周期为

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - \frac{1}{4} + i \frac{3}{4} :

无解

sin (x) = - \frac{1}{4} + i \frac{3}{4}

无解

sin (x) = - \frac{1}{4} - i \frac{3}{4} :

无解

sin (x) = - \frac{1}{4} - i \frac{3}{4}

无解

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

流行的例子

4sin^2(x)=8sin^2(x/2)

4 sin^{2} (x) = 8 sin^{2} (\frac{x}{2})

cos (\frac{π}{2} - x) = 0

1sin(45)=2.42sin(r)

1 sin (4 5^{\circ}) = 2.42 sin (r)

sec (x) = 4 sin (x)

sinh (nπ) = 0