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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 6 0^{\circ} n

求解步骤

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

两边减去

sin (2 x) - cos (2 x)

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

使用三角恒等式改写

cos (2 x) - cos (4 x) - sin (2 x) + sin (4 x)

使用和差化积恒等式:

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

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

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

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

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

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

同类项相加:

4 x - 2 x = 2 x

= \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

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

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

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

同类项相加:

4 x + 2 x = 6 x

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 x

= 2 sin (x) cos (3 x)

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

化简

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

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

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

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

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

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

同类项相加:

2 x + 4 x = 6 x

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 x

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

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

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

同类项相加:

2 x - 4 x = - 2 x

= \frac{- 2 x}{2}

使用分式法则:

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

= - \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= - x

= 2 sin (3 x) sin (- x)

使用负角恒等式:

sin (- x) = - sin (x)

= 2 (- sin (x)) sin (3 x)

= 2 cos (3 x) sin (x) - (- 2 sin (x) sin (3 x))

使用法则

- (- a) = a

= 2 cos (3 x) sin (x) + 2 sin (x) sin (3 x)

= 2 cos (3 x) sin (x) + 2 sin (x) sin (3 x)

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

分解

2 cos (3 x) sin (x) + 2 sin (3 x) sin (x) : 2 sin (x) (cos (3 x) + sin (3 x))

2 cos (3 x) sin (x) + 2 sin (3 x) sin (x)

改写为

= 2 sin (x) cos (3 x) + 2 sin (x) sin (3 x)

因式分解出通项

2 sin (x)

= 2 sin (x) (cos (3 x) + sin (3 x))

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

分别求解每个部分

sin (x) = 0 or cos (3 x) + sin (3 x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

cos (3 x) + sin (3 x) = 0

使用三角恒等式改写

cos (3 x) + sin (3 x) = 0

在两边除以

cos (3 x), cos (3 x) \neq = 0

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

化简

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

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 + tan (3 x) = 0

1 + tan (3 x) = 0

将

1

到右边

1 + tan (3 x) = 0

两边减去

1

1 + tan (3 x) - 1 = 0 - 1

化简

tan (3 x) = - 1

tan (3 x) = - 1

tan (3 x) = - 1

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

3 x = \frac{3 π}{4} + πn

3 x = \frac{3 π}{4} + πn

解

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

3 x = \frac{3 π}{4} + πn

两边除以

3

3 x = \frac{3 π}{4} + πn

两边除以

3

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

化简

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

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{\frac{3 π}{4}}{3} + \frac{πn}{3} : \frac{π}{4} + \frac{πn}{3}

\frac{\frac{3 π}{4}}{3} + \frac{πn}{3}

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

使用分式法则:

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

= \frac{3 π}{4 \cdot 3}

数字相乘:

4 \cdot 3 = 12

= \frac{3 π}{12}

约分:

3

= \frac{π}{4}

= \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

合并所有解

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

作图

流行的例子

2 \cdot sin^{2} (x) - 1 = 0

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

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

tan(θ)= 220/370

tan (θ) = \frac{220}{370}

2sin(3x)*sin(x)=1

2 sin (3 x) \cdot sin (x) = 1

arcsin(6x)+arcsin(6sqrt(3x))=-pi/2

arcsin (6 x) + arcsin (6 3 x) = - \frac{π}{2}