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受欢迎的三角函数 >

cos^2(x)=cos^2(x/2)

解答

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

解答

x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

+1

度数

x = 12 0^{\circ} + 48 0^{\circ} n, x = 36 0^{\circ} + 48 0^{\circ} n, x = 0^{\circ} + 48 0^{\circ} n, x = 24 0^{\circ} + 48 0^{\circ} n

求解步骤

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

两边减去

cos^{2} (\frac{x}{2})

cos^{2} (x) - cos^{2} (\frac{x}{2}) = 0

分解

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

cos^{2} (x) - cos^{2} (\frac{x}{2})

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

分别求解每个部分

cos (x) + cos (\frac{x}{2}) = 0 or cos (x) - cos (\frac{x}{2}) = 0

cos (x) + cos (\frac{x}{2}) = 0 : x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}

cos (x) + cos (\frac{x}{2}) = 0

使用三角恒等式改写

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

化简

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

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

\frac{\frac{x}{2} + x}{2} = \frac{3 x}{4}

\frac{\frac{x}{2} + x}{2}

化简

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

\frac{x}{2} + x

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x}{2} + \frac{x \cdot 2}{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{x + x \cdot 2}{2}

同类项相加:

x + 2 x = 3 x

= \frac{3 x}{2}

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

使用分式法则:

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

= \frac{3 x}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{3 x}{4}

= 2 cos (\frac{3 x}{4}) cos (\frac{\frac{x}{2} - x}{2})

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

\frac{\frac{x}{2} - x}{2}

化简

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

\frac{x}{2} - x

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x}{2} - \frac{x \cdot 2}{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{x - x \cdot 2}{2}

同类项相加:

x - 2 x = - x

= \frac{- x}{2}

使用分式法则:

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

= - \frac{x}{2}

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

使用分式法则:

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

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

使用分式法则:

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

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

= - \frac{x}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= - \frac{x}{4}

= 2 cos (\frac{3 x}{4}) cos (- \frac{x}{4})

使用负角恒等式:

cos (- x) = cos (x)

= 2 cos (\frac{x}{4}) cos (\frac{3 x}{4})

= 2 cos (\frac{x}{4}) cos (\frac{3 x}{4})

2 cos (\frac{3 x}{4}) cos (\frac{x}{4}) = 0

分别求解每个部分

cos (\frac{3 x}{4}) = 0 or cos (\frac{x}{4}) = 0

cos (\frac{3 x}{4}) = 0 : x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}

cos (\frac{3 x}{4}) = 0

cos (\frac{3 x}{4}) = 0

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}

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

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

解

\frac{3 x}{4} = \frac{π}{2} + 2 πn : x = \frac{2 π}{3} + \frac{8 πn}{3}

\frac{3 x}{4} = \frac{π}{2} + 2 πn

在两边乘以

4

\frac{3 x}{4} = \frac{π}{2} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 \cdot 3 x}{4} : 3 x

\frac{4 \cdot 3 x}{4}

数字相乘:

4 \cdot 3 = 12

= \frac{12 x}{4}

数字相除:

\frac{12}{4} = 3

= 3 x

化简

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

分式相乘:

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

= \frac{π 4}{2}

数字相除:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

3 x = 2 π + 8 πn

3 x = 2 π + 8 πn

3 x = 2 π + 8 πn

两边除以

3

3 x = 2 π + 8 πn

两边除以

3

\frac{3 x}{3} = \frac{2 π}{3} + \frac{8 πn}{3}

化简

x = \frac{2 π}{3} + \frac{8 πn}{3}

x = \frac{2 π}{3} + \frac{8 πn}{3}

解

\frac{3 x}{4} = \frac{3 π}{2} + 2 πn : x = 2 π + \frac{8 πn}{3}

\frac{3 x}{4} = \frac{3 π}{2} + 2 πn

在两边乘以

4

\frac{3 x}{4} = \frac{3 π}{2} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 \cdot 3 x}{4} : 3 x

\frac{4 \cdot 3 x}{4}

数字相乘:

4 \cdot 3 = 12

= \frac{12 x}{4}

数字相除:

\frac{12}{4} = 3

= 3 x

化简

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

分式相乘:

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

= \frac{3 π 4}{2}

数字相乘:

3 \cdot 4 = 12

= \frac{12 π}{2}

数字相除:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

3 x = 6 π + 8 πn

3 x = 6 π + 8 πn

3 x = 6 π + 8 πn

两边除以

3

3 x = 6 π + 8 πn

两边除以

3

\frac{3 x}{3} = \frac{6 π}{3} + \frac{8 πn}{3}

化简

x = 2 π + \frac{8 πn}{3}

x = 2 π + \frac{8 πn}{3}

x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}

cos (\frac{x}{4}) = 0 : x = 2 π + 8 πn, x = 6 π + 8 πn

cos (\frac{x}{4}) = 0

cos (\frac{x}{4}) = 0

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}

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

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

解

\frac{x}{4} = \frac{π}{2} + 2 πn : x = 2 π + 8 πn

\frac{x}{4} = \frac{π}{2} + 2 πn

在两边乘以

4

\frac{x}{4} = \frac{π}{2} + 2 πn

在两边乘以

4

\frac{4 x}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

化简

\frac{4 x}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

分式相乘:

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

= \frac{π 4}{2}

数字相除:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

x = 2 π + 8 πn

x = 2 π + 8 πn

x = 2 π + 8 πn

解

\frac{x}{4} = \frac{3 π}{2} + 2 πn : x = 6 π + 8 πn

\frac{x}{4} = \frac{3 π}{2} + 2 πn

在两边乘以

4

\frac{x}{4} = \frac{3 π}{2} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

分式相乘:

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

= \frac{3 π 4}{2}

数字相乘:

3 \cdot 4 = 12

= \frac{12 π}{2}

数字相除:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

x = 6 π + 8 πn

x = 6 π + 8 πn

x = 6 π + 8 πn

x = 2 π + 8 πn, x = 6 π + 8 πn

合并所有解

x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 2 π + 8 πn, x = 6 π + 8 πn

合并重叠的区间

x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}

cos (x) - cos (\frac{x}{2}) = 0 : x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

cos (x) - cos (\frac{x}{2}) = 0

使用三角恒等式改写

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

化简

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

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

\frac{x + \frac{x}{2}}{2} = \frac{3 x}{4}

\frac{x + \frac{x}{2}}{2}

化简

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

x + \frac{x}{2}

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x \cdot 2}{2} + \frac{x}{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{x \cdot 2 + x}{2}

同类项相加:

2 x + x = 3 x

= \frac{3 x}{2}

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

使用分式法则:

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

= \frac{3 x}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{3 x}{4}

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

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

\frac{x - \frac{x}{2}}{2}

化简

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

x - \frac{x}{2}

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x \cdot 2}{2} - \frac{x}{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{x \cdot 2 - x}{2}

同类项相加:

2 x - x = x

= \frac{x}{2}

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

使用分式法则:

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

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

数字相乘:

2 \cdot 2 = 4

= \frac{x}{4}

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

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

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

分别求解每个部分

sin (\frac{3 x}{4}) = 0 or sin (\frac{x}{4}) = 0

sin (\frac{3 x}{4}) = 0 : x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (\frac{3 x}{4}) = 0

sin (\frac{3 x}{4}) = 0

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}

\frac{3 x}{4} = 0 + 2 πn, \frac{3 x}{4} = π + 2 πn

\frac{3 x}{4} = 0 + 2 πn, \frac{3 x}{4} = π + 2 πn

解

\frac{3 x}{4} = 0 + 2 πn : x = \frac{8 πn}{3}

\frac{3 x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x}{4} = 2 πn

在两边乘以

4

\frac{3 x}{4} = 2 πn

在两边乘以

4

\frac{4 \cdot 3 x}{4} = 4 \cdot 2 πn

化简

3 x = 8 πn

3 x = 8 πn

两边除以

3

3 x = 8 πn

两边除以

3

\frac{3 x}{3} = \frac{8 πn}{3}

化简

x = \frac{8 πn}{3}

x = \frac{8 πn}{3}

解

\frac{3 x}{4} = π + 2 πn : x = \frac{4 π}{3} + \frac{8 πn}{3}

\frac{3 x}{4} = π + 2 πn

在两边乘以

4

\frac{3 x}{4} = π + 2 πn

在两边乘以

4

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

化简

3 x = 4 π + 8 πn

3 x = 4 π + 8 πn

两边除以

3

3 x = 4 π + 8 πn

两边除以

3

\frac{3 x}{3} = \frac{4 π}{3} + \frac{8 πn}{3}

化简

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (\frac{x}{4}) = 0 : x = 8 πn, x = 4 π + 8 πn

sin (\frac{x}{4}) = 0

sin (\frac{x}{4}) = 0

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}

\frac{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

\frac{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

解

\frac{x}{4} = 0 + 2 πn : x = 8 πn

\frac{x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{4} = 2 πn

在两边乘以

4

\frac{x}{4} = 2 πn

在两边乘以

4

\frac{4 x}{4} = 4 \cdot 2 πn

化简

x = 8 πn

x = 8 πn

解

\frac{x}{4} = π + 2 πn : x = 4 π + 8 πn

\frac{x}{4} = π + 2 πn

在两边乘以

4

\frac{x}{4} = π + 2 πn

在两边乘以

4

\frac{4 x}{4} = 4 π + 4 \cdot 2 πn

化简

x = 4 π + 8 πn

x = 4 π + 8 πn

x = 8 πn, x = 4 π + 8 πn

合并所有解

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

合并重叠的区间

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

合并所有解

x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

作图

流行的例子

csc^2(θ)-csc(θ)-20=0

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sec^2(θ)+7sec(θ)+6=0

sec^{2} (θ) + 7 sec (θ) + 6 = 0

cos (x) = \frac{3}{8}

- 1 = cos (x)

cos(2x)-3sin(x)+1=0

cos (2 x) - 3 sin (x) + 1 = 0