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受欢迎的三角函数 >

sin(x/2)=sin(2x)

解答

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

解答

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

+1

度数

x = 7 2^{\circ} + 28 8^{\circ} n, x = 21 6^{\circ} + 28 8^{\circ} n, x = 0^{\circ} + 48 0^{\circ} n, x = 24 0^{\circ} + 48 0^{\circ} n

求解步骤

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

两边减去

sin (2 x)

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

使用三角恒等式改写

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

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

化简

2 sin (\frac{\frac{x}{2} - 2 x}{2}) cos (\frac{\frac{x}{2} + 2 x}{2}) : - 2 cos (\frac{5 x}{4}) sin (\frac{3 x}{4})

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

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

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

化简

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

\frac{x}{2} - 2 x

将项转换为分式:

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

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

x - 2 x \cdot 2 = - 3 x

x - 2 x \cdot 2

数字相乘:

2 \cdot 2 = 4

= x - 4 x

同类项相加:

x - 4 x = - 3 x

= - 3 x

= \frac{- 3 x}{2}

使用分式法则:

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

= - \frac{3 x}{2}

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

使用分式法则:

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

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

使用分式法则:

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

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

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

数字相乘:

2 \cdot 2 = 4

= - \frac{3 x}{4}

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

使用负角恒等式:

sin (- x) = - sin (x)

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

去除括号:

(- a) = - a

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

\frac{\frac{x}{2} + 2 x}{2} = \frac{5 x}{4}

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

化简

\frac{x}{2} + 2 x : \frac{5 x}{2}

\frac{x}{2} + 2 x

将项转换为分式:

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

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

x + 2 x \cdot 2 = 5 x

x + 2 x \cdot 2

数字相乘:

2 \cdot 2 = 4

= x + 4 x

同类项相加:

x + 4 x = 5 x

= 5 x

= \frac{5 x}{2}

= \frac{\frac{5 x}{2}}{2}

使用分式法则:

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

= \frac{5 x}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{5 x}{4}

= - 2 cos (\frac{5 x}{4}) sin (\frac{3 x}{4})

= - 2 cos (\frac{5 x}{4}) sin (\frac{3 x}{4})

- 2 cos (\frac{5 x}{4}) sin (\frac{3 x}{4}) = 0

分别求解每个部分

cos (\frac{5 x}{4}) = 0 or sin (\frac{3 x}{4}) = 0

cos (\frac{5 x}{4}) = 0 : x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}

cos (\frac{5 x}{4}) = 0

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

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

解

\frac{5 x}{4} = \frac{π}{2} + 2 πn : x = \frac{2 π}{5} + \frac{8 πn}{5}

\frac{5 x}{4} = \frac{π}{2} + 2 πn

在两边乘以

4

\frac{5 x}{4} = \frac{π}{2} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 \cdot 5 x}{4} : 5 x

\frac{4 \cdot 5 x}{4}

数字相乘:

4 \cdot 5 = 20

= \frac{20 x}{4}

数字相除:

\frac{20}{4} = 5

= 5 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

5 x = 2 π + 8 πn

5 x = 2 π + 8 πn

5 x = 2 π + 8 πn

两边除以

5

5 x = 2 π + 8 πn

两边除以

5

\frac{5 x}{5} = \frac{2 π}{5} + \frac{8 πn}{5}

化简

x = \frac{2 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}

解

\frac{5 x}{4} = \frac{3 π}{2} + 2 πn : x = \frac{6 π}{5} + \frac{8 πn}{5}

\frac{5 x}{4} = \frac{3 π}{2} + 2 πn

在两边乘以

4

\frac{5 x}{4} = \frac{3 π}{2} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 \cdot 5 x}{4} : 5 x

\frac{4 \cdot 5 x}{4}

数字相乘:

4 \cdot 5 = 20

= \frac{20 x}{4}

数字相除:

\frac{20}{4} = 5

= 5 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

5 x = 6 π + 8 πn

5 x = 6 π + 8 πn

5 x = 6 π + 8 πn

两边除以

5

5 x = 6 π + 8 πn

两边除以

5

\frac{5 x}{5} = \frac{6 π}{5} + \frac{8 πn}{5}

化简

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}

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}

合并所有解

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

作图

流行的例子

cos (x) = cos (1)

sin^{2} (2 θ) = 0

solvefor y,x=2sec(y)

so l v e f or y, x = 2 sec (y)

tan (π + c) = 1

8sin^2(x)=8cos^2(x)

8 sin^{2} (x) = 8 cos^{2} (x)