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受欢迎的三角函数 >

cos(pi/3+x)+cos(pi/3-x)= 1/2

解答

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

解答

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

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

cos (\frac{π}{3} + x) + cos (\frac{π}{3} - x)

使用和差化积恒等式:

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

= 2 cos (\frac{\frac{π}{3} + x + \frac{π}{3} - x}{2}) cos (\frac{\frac{π}{3} + x - ( \frac{π}{3} - x )}{2})

化简

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

2 cos (\frac{\frac{π}{3} + x + \frac{π}{3} - x}{2}) cos (\frac{\frac{π}{3} + x - ( \frac{π}{3} - x )}{2})

\frac{\frac{π}{3} + x + \frac{π}{3} - x}{2} = \frac{π}{3}

\frac{\frac{π}{3} + x + \frac{π}{3} - x}{2}

合并分式

\frac{π}{3} + \frac{π}{3} : \frac{2 π}{3}

使用法则

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

= \frac{π + π}{3}

同类项相加:

π + π = 2 π

= \frac{2 π}{3}

= \frac{\frac{2 π}{3} + x - x}{2}

同类项相加:

x - x = 0

= \frac{\frac{2 π}{3}}{2}

使用分式法则:

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

= \frac{2 π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{2 π}{6}

约分:

2

= \frac{π}{3}

= 2 cos (\frac{π}{3}) cos (\frac{x - ( - x + \frac{π}{3} ) + \frac{π}{3}}{2})

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

\frac{\frac{π}{3} + x - ( \frac{π}{3} - x )}{2}

化简

\frac{π}{3} + x - (\frac{π}{3} - x) : 2 x

\frac{π}{3} + x - (\frac{π}{3} - x)

将项转换为分式:

x = \frac{x 3}{3}, (- x + \frac{π}{3}) = \frac{( \frac{π}{3} - x ) 3}{3}

= \frac{π}{3} + \frac{x \cdot 3}{3} - \frac{( \frac{π}{3} - x ) \cdot 3}{3}

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{π + x \cdot 3 - ( \frac{π}{3} - x ) \cdot 3}{3}

乘开

π + x \cdot 3 - (\frac{π}{3} - x) \cdot 3 : 6 x

π + x \cdot 3 - (\frac{π}{3} - x) \cdot 3

= π + 3 x - 3 (\frac{π}{3} - x)

乘开

- 3 (\frac{π}{3} - x) : - π + 3 x

- 3 (\frac{π}{3} - x)

使用分配律:

a (b - c) = ab - a c

a = - 3, b = \frac{π}{3}, c = x

= - 3 \cdot \frac{π}{3} - (- 3) x

使用加减运算法则

- (- a) = a

= - 3 \cdot \frac{π}{3} + 3 x

3 \cdot \frac{π}{3} = π

3 \cdot \frac{π}{3}

分式相乘:

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

= \frac{π 3}{3}

约分:

3

= π

= - π + 3 x

= π + x \cdot 3 - π + 3 x

化简

π + x \cdot 3 - π + 3 x : 6 x

π + x \cdot 3 - π + 3 x

对同类项分组

= 3 x + 3 x + π - π

同类项相加:

3 x + 3 x = 6 x

= 6 x + π - π

同类项相加:

π - π = 0

= 6 x

= 6 x

= \frac{6 x}{3}

数字相除:

\frac{6}{3} = 2

= 2 x

= \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

= 2 cos (\frac{π}{3}) cos (x)

化简

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

使用以下普通恒等式:

cos (\frac{π}{3}) = \frac{1}{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{1}{2}

= 2 \cdot \frac{1}{2} cos (x)

分式相乘:

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

= \frac{1 \cdot 2}{2} cos (x)

约分:

2

= cos (x) \cdot 1

乘以:

cos (x) \cdot 1 = cos (x)

= cos (x)

= cos (x)

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

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

的通解

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}

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

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

作图

流行的例子

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