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受欢迎的三角函数 >

sqrt(2)sin(2x)=sin(4x)

解答

2 sin (2 x) = sin (4 x)

解答

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{π + 8 πn}{8}, x = \frac{7 π + 8 πn}{8}

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 22. 5^{\circ} + 18 0^{\circ} n, x = 157. 5^{\circ} + 18 0^{\circ} n

求解步骤

2 sin (2 x) = sin (4 x)

两边减去

sin (4 x)

2 sin (2 x) - sin (4 x) = 0

令

: u = 2 x

2 sin (u) - sin (2 u) = 0

使用三角恒等式改写

- sin (2 u) + sin (u) 2

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= - 2 sin (u) cos (u) + 2 sin (u)

sin (u) 2 - 2 cos (u) sin (u) = 0

分解

sin (u) 2 - 2 cos (u) sin (u) : sin (u) (2 - 2 cos (u))

sin (u) 2 - 2 cos (u) sin (u)

因式分解出通项

sin (u)

= sin (u) (2 - 2 cos (u))

sin (u) (2 - 2 cos (u)) = 0

分别求解每个部分

sin (u) = 0 or 2 - 2 cos (u) = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

解

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 - 2 cos (u) = 0 : u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

2 - 2 cos (u) = 0

将

2

到右边

2 - 2 cos (u) = 0

两边减去

2

2 - 2 cos (u) - 2 = 0 - 2

化简

- 2 cos (u) = - 2

- 2 cos (u) = - 2

两边除以

- 2

- 2 cos (u) = - 2

两边除以

- 2

\frac{- 2 cos ( u )}{- 2} = \frac{- 2}{- 2}

化简

cos (u) = \frac{2}{2}

cos (u) = \frac{2}{2}

cos (u) = \frac{2}{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}

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

合并所有解

u = 2 πn, u = π + 2 πn, u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

u = 2 x

代回

2 x = 2 πn : x = πn

2 x = 2 πn

两边除以

2

2 x = 2 πn

两边除以

2

\frac{2 x}{2} = \frac{2 πn}{2}

化简

x = πn

x = πn

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

2 x = π + 2 πn

两边除以

2

2 x = π + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

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

\frac{π}{2} + \frac{2 πn}{2}

使用法则

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

= \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

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

2 x = \frac{π}{4} + 2 πn

两边除以

2

2 x = \frac{π}{4} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

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

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

使用法则

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

= \frac{\frac{π}{4} + 2 πn}{2}

化简

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

\frac{π}{4} + 2 πn

将项转换为分式:

2 πn = \frac{2 πn 4}{4}

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

数字相乘:

2 \cdot 4 = 8

= \frac{π + 8 πn}{4}

= \frac{\frac{π + 8 πn}{4}}{2}

使用分式法则:

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

= \frac{π + 8 πn}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π + 8 πn}{8}

2 x = \frac{7 π}{4} + 2 πn

两边除以

2

2 x = \frac{7 π}{4} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π + 8 πn}{8}

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

使用法则

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

= \frac{\frac{7 π}{4} + 2 πn}{2}

化简

\frac{7 π}{4} + 2 πn : \frac{7 π + 8 πn}{4}

\frac{7 π}{4} + 2 πn

将项转换为分式:

2 πn = \frac{2 πn 4}{4}

= \frac{7 π}{4} + \frac{2 πn \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{7 π + 2 πn \cdot 4}{4}

数字相乘:

2 \cdot 4 = 8

= \frac{7 π + 8 πn}{4}

= \frac{\frac{7 π + 8 πn}{4}}{2}

使用分式法则:

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

= \frac{7 π + 8 πn}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{π + 8 πn}{8}, x = \frac{7 π + 8 πn}{8}

作图

流行的例子

sin^2(x)-sin(x)-6=0

sin^{2} (x) - sin (x) - 6 = 0

cos (z) = sin (z)

4sin^2(x)=(csc^2(x))/4

4 sin^{2} (x) = \frac{csc ^{2} ( x )}{4}

2 sin (3 x) = 2

sin(x)=(sqrt(2))/3

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