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受欢迎的三角函数 >

sin(x+pi/4)+sin(x-pi/4)+(sqrt(6))/2 =0

解答

sin (x + \frac{π}{4}) + sin (x - \frac{π}{4}) + \frac{6}{2} = 0

解答

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

sin (x + \frac{π}{4}) + sin (x - \frac{π}{4}) + \frac{6}{2} = 0

使用三角恒等式改写

sin (x + \frac{π}{4}) + sin (x - \frac{π}{4}) + \frac{6}{2}

使用和差化积恒等式:

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

= \frac{6}{2} + 2 sin (\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2}) cos (\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2})

化简

\frac{6}{2} + 2 sin (\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2}) cos (\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2}) : \frac{3}{2} + 2 sin (x)

\frac{6}{2} + 2 sin (\frac{x + \frac{π}{4} + x - \frac{π}{4}}{2}) cos (\frac{x + \frac{π}{4} - ( x - \frac{π}{4} )}{2})

\frac{6}{2} = \frac{3}{2}

\frac{6}{2}

分解

6 : 23

因式分解

6 = 2 \cdot 3

= 2 \cdot 3

使用根式运算法则:

n ab = n a n b

= 23

= \frac{2 3}{2}

消掉

\frac{2 3}{2} : \frac{3}{2}

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

数字相减:

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

= \frac{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{3}{2}

= \frac{3}{2}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{2}

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

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

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

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

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

x + \frac{π}{4} + x - \frac{π}{4}

对同类项分组

= x + x + \frac{π}{4} - \frac{π}{4}

同类项相加:

x + x = 2 x

= 2 x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= 2 x

= \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

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

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

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

化简

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

x + \frac{π}{4} - (x - \frac{π}{4})

将项转换为分式:

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

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

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{x \cdot 4 + π - ( x - \frac{π}{4} ) \cdot 4}{4}

乘开

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

x \cdot 4 + π - (x - \frac{π}{4}) \cdot 4

= 4 x + π - 4 (x - \frac{π}{4})

乘开

- 4 (x - \frac{π}{4}) : - 4 x + π

- 4 (x - \frac{π}{4})

使用分配律:

a (b - c) = ab - a c

a = - 4, b = x, c = \frac{π}{4}

= - 4 x - (- 4) \frac{π}{4}

使用加减运算法则

- (- a) = a

= - 4 x + 4 \cdot \frac{π}{4}

4 \cdot \frac{π}{4} = π

4 \cdot \frac{π}{4}

分式相乘:

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

= \frac{π 4}{4}

约分:

4

= π

= - 4 x + π

= x \cdot 4 + π - 4 x + π

化简

x \cdot 4 + π - 4 x + π : 2 π

x \cdot 4 + π - 4 x + π

对同类项分组

= 4 x - 4 x + π + π

同类项相加:

4 x - 4 x = 0

= π + π

同类项相加:

π + π = 2 π

= 2 π

= 2 π

= \frac{2 π}{4}

约分:

2

= \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 cos (\frac{π}{4}) sin (x)

化简

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

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

分式相乘:

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

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

约分:

2

= 2 sin (x)

= \frac{3}{2} + 2 sin (x)

= \frac{3}{2} + 2 sin (x)

\frac{3}{2} + 2 sin (x) = 0

将

\frac{3}{2}

到右边

\frac{3}{2} + 2 sin (x) = 0

两边减去

\frac{3}{2}

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

化简

2 sin (x) = - \frac{3}{2}

2 sin (x) = - \frac{3}{2}

两边除以

2

2 sin (x) = - \frac{3}{2}

两边除以

2

\frac{2 sin ( x )}{2} = \frac{- \frac{3}{2}}{2}

化简

\frac{2 sin ( x )}{2} = \frac{- \frac{3}{2}}{2}

化简

\frac{2 sin ( x )}{2} : sin (x)

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

约分:

2

= sin (x)

化简

\frac{- \frac{3}{2}}{2} : - \frac{3}{2}

\frac{- \frac{3}{2}}{2}

使用分式法则:

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

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

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

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}{2}

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

使用分式法则:

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

= - \frac{3}{2 2}

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= - \frac{3}{2}

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

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