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受欢迎的三角函数 >

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

解答

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

解答

x = 0.54408 \dots + 2 πn - \frac{π}{4}, x = π - 0.54408 \dots + 2 πn - \frac{π}{4}

+1

度数

x = - 13.82604 \dots^{\circ} + 36 0^{\circ} n, x = 103.82604 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

3 - sin (x)

1 + cos (x) - 3 + sin (x) = 0

使用三角恒等式改写

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

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

sin (x) + cos (x)

改写为

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

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{1}{2}

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{1}{2}

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

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (x + \frac{π}{4})

= 1 - 3 + 2 sin (x + \frac{π}{4})

1 - 3 + 2 sin (x + \frac{π}{4}) = 0

将

1

到右边

1 - 3 + 2 sin (x + \frac{π}{4}) = 0

两边减去

1

1 - 3 + 2 sin (x + \frac{π}{4}) - 1 = 0 - 1

化简

- 3 + 2 sin (x + \frac{π}{4}) = - 1

- 3 + 2 sin (x + \frac{π}{4}) = - 1

将

3

到右边

- 3 + 2 sin (x + \frac{π}{4}) = - 1

两边加上

3

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

化简

2 sin (x + \frac{π}{4}) = - 1 + 3

2 sin (x + \frac{π}{4}) = - 1 + 3

两边除以

2

2 sin (x + \frac{π}{4}) = - 1 + 3

两边除以

2

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

化简

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

化简

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

\frac{2 sin ( x + \frac{π}{4} )}{2}

约分:

2

= sin (x + \frac{π}{4})

化简

- \frac{1}{2} + \frac{3}{2} : \frac{2 ( - 1 + 3 )}{2}

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

使用法则

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

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

乘以共轭根式

\frac{2}{2}

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

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

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

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

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

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

使用反三角函数性质

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

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

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn, x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn, x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

解

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn : x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

化简

arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn : arcsin (\frac{- 1 + 3}{2}) + 2 πn

arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

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

\frac{2 ( - 1 + 3 )}{2}

使用根式运算法则:

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

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

= \frac{2 ^{\frac{1}{2}} ( 3 - 1 )}{2}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

使用根式运算法则:

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

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

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

= arcsin (\frac{3 - 1}{2}) + 2 πn

x + \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn

将

\frac{π}{4}

到右边

x + \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn

两边减去

\frac{π}{4}

x + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

化简

x + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

化简

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= x

化简

arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4} : arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

= arcsin (\frac{2 ( 3 - 1 )}{2}) + 2 πn - \frac{π}{4}

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

\frac{2 ( - 1 + 3 )}{2}

使用根式运算法则:

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

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

= \frac{2 ^{\frac{1}{2}} ( 3 - 1 )}{2}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

使用根式运算法则:

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

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

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

= arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

无法进一步化简

= arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

解

x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn : x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

化简

π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn : π - arcsin (\frac{- 1 + 3}{2}) + 2 πn

π - arcsin (\frac{2 ( - 1 + 3 )}{2}) + 2 πn

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

\frac{2 ( - 1 + 3 )}{2}

使用根式运算法则:

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

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

= \frac{2 ^{\frac{1}{2}} ( 3 - 1 )}{2}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

使用根式运算法则:

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

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

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

= π - arcsin (\frac{3 - 1}{2}) + 2 πn

x + \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn

将

\frac{π}{4}

到右边

x + \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn

两边减去

\frac{π}{4}

x + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

化简

x + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

化简

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= x

化简

π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4} : π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

= π - arcsin (\frac{2 ( 3 - 1 )}{2}) + 2 πn - \frac{π}{4}

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

\frac{2 ( - 1 + 3 )}{2}

使用根式运算法则:

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

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

= \frac{2 ^{\frac{1}{2}} ( 3 - 1 )}{2}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3}{2 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

使用根式运算法则:

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

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

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

= π - arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

无法进一步化简

= π - arcsin (\frac{3 - 1}{2}) + 2 πn - \frac{π}{4}

x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}, x = π - arcsin (\frac{- 1 + 3}{2}) + 2 πn - \frac{π}{4}

以小数形式表示解

x = 0.54408 \dots + 2 πn - \frac{π}{4}, x = π - 0.54408 \dots + 2 πn - \frac{π}{4}

作图

流行的例子

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