zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

cos(x)-sin(x)=2

解答

cos (x) - sin (x) = 2

解答

x \in R 无解

求解步骤

cos (x) - sin (x) = 2

两边减去

2

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

使用三角恒等式改写

- 2 + cos (x) - sin (x)

利用以下特性:

sin (x) = cos (\frac{π}{2} - x)

= - 2 + cos (x) - cos (\frac{π}{2} - x)

使用和差化积恒等式:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 - 2 sin (\frac{x + \frac{π}{2} - x}{2}) sin (\frac{x - ( \frac{π}{2} - x )}{2})

2 sin (\frac{x + \frac{π}{2} - x}{2}) sin (\frac{x - ( \frac{π}{2} - x )}{2}) = 2 sin (\frac{4 x - π}{4})

2 sin (\frac{x + \frac{π}{2} - x}{2}) sin (\frac{x - ( \frac{π}{2} - x )}{2})

\frac{x + \frac{π}{2} - x}{2} = \frac{π}{4}

\frac{x + \frac{π}{2} - x}{2}

x + \frac{π}{2} - x = \frac{π}{2}

x + \frac{π}{2} - x

对同类项分组

= x - x + \frac{π}{2}

同类项相加:

x - x = 0

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

使用分式法则:

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

= \frac{π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{π}{4}

= 2 sin (\frac{π}{4}) sin (\frac{x - ( - x + \frac{π}{2} )}{2})

\frac{x - ( \frac{π}{2} - x )}{2} = \frac{4 x - π}{4}

\frac{x - ( \frac{π}{2} - x )}{2}

乘开

x - (\frac{π}{2} - x) : 2 x - \frac{π}{2}

x - (\frac{π}{2} - x)

- (\frac{π}{2} - x) : - \frac{π}{2} + x

- (\frac{π}{2} - x)

打开括号

= - (\frac{π}{2}) - (- x)

使用加减运算法则

- (- a) = a, - (a) = - a

= - \frac{π}{2} + x

= x - \frac{π}{2} + x

化简

x - \frac{π}{2} + x : 2 x - \frac{π}{2}

x - \frac{π}{2} + x

对同类项分组

= x + x - \frac{π}{2}

同类项相加:

x + x = 2 x

= 2 x - \frac{π}{2}

= 2 x - \frac{π}{2}

= \frac{2 x - \frac{π}{2}}{2}

化简

2 x - \frac{π}{2} : \frac{4 x - π}{2}

2 x - \frac{π}{2}

将项转换为分式:

2 x = \frac{2 x 2}{2}

= \frac{2 x \cdot 2}{2} - \frac{π}{2}

因为分母相等，所以合并分式:

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

= \frac{2 x \cdot 2 - π}{2}

数字相乘:

2 \cdot 2 = 4

= \frac{4 x - π}{2}

= \frac{\frac{4 x - π}{2}}{2}

使用分式法则:

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

= \frac{4 x - π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{4 x - π}{4}

= 2 sin (\frac{π}{4}) sin (\frac{4 x - π}{4})

化简

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

= 2 \cdot \frac{2}{2} sin (\frac{4 x - π}{4})

分式相乘:

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

= \frac{2 \cdot 2 sin ( \frac{4 x - π}{4} )}{2}

约分:

2

= 2 sin (\frac{4 x - π}{4})

= - 2 - 2 sin (\frac{4 x - π}{4})

- 2 - 2 sin (\frac{4 x - π}{4}) = 0

将

2

到右边

- 2 - 2 sin (\frac{4 x - π}{4}) = 0

两边加上

2

- 2 - 2 sin (\frac{4 x - π}{4}) + 2 = 0 + 2

化简

- 2 sin (\frac{4 x - π}{4}) = 2

- 2 sin (\frac{4 x - π}{4}) = 2

两边除以

- 2

- 2 sin (\frac{4 x - π}{4}) = 2

两边除以

- 2

\frac{- 2 sin ( \frac{4 x - π}{4} )}{- 2} = \frac{2}{- 2}

化简

\frac{- 2 sin ( \frac{4 x - π}{4} )}{- 2} = \frac{2}{- 2}

化简

\frac{- 2 sin ( \frac{4 x - π}{4} )}{- 2} : sin (\frac{4 x - π}{4})

\frac{- 2 sin ( \frac{4 x - π}{4} )}{- 2}

使用分式法则:

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

= \frac{2 sin ( \frac{4 x - π}{4} )}{2}

约分:

2

= sin (\frac{4 x - π}{4})

化简

\frac{2}{- 2} : - 2

\frac{2}{- 2}

使用分式法则:

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

= - \frac{2}{2}

使用根式运算法则:

n a = a^{\frac{1}{n}}

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

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

使用指数法则:

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

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

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

数字相减:

1 - \frac{1}{2} = \frac{1}{2}

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

使用根式运算法则:

a^{\frac{1}{n}} = n a

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

= - 2

sin (\frac{4 x - π}{4}) = - 2

sin (\frac{4 x - π}{4}) = - 2

sin (\frac{4 x - π}{4}) = - 2

- 1 \leq sin (x) \leq 1

x \in R 无解

作图

流行的例子

cot(x)+sqrt(2)cos(x)=0

cot (x) + 2 cos (x) = 0

3cos(x)-sin(x)=1

3 cos (x) - sin (x) = 1

3 sin (θ) - 1 = 0

tan (x) = \frac{40}{9}

cos((3x)/2)= 1/2 ,-pi/2 <= x<= pi/2

cos (\frac{3 x}{2}) = \frac{1}{2}, - \frac{π}{2} \leq x \leq \frac{π}{2}