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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 67. 5^{\circ} + 9 0^{\circ} n

求解步骤

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

两边减去

- 2 sin (2 x) sin (x)

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

使用三角恒等式改写

sin (3 x) - sin (x) + 2 sin (2 x) sin (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})

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

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

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

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

\frac{3 x - x}{2}

同类项相加:

3 x - x = 2 x

= \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

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

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

\frac{3 x + x}{2}

同类项相加:

3 x + x = 4 x

= \frac{4 x}{2}

数字相除:

\frac{4}{2} = 2

= 2 x

= 2 sin (x) cos (2 x)

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

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

分解

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

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

改写为

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

因式分解出通项

2 sin (x)

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

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

分别求解每个部分

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

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

使用三角恒等式改写

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

在两边除以

cos (2 x), cos (2 x) \neq = 0

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

化简

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

使用基本三角恒等式:

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

1 + tan (2 x) = 0

1 + tan (2 x) = 0

将

1

到右边

1 + tan (2 x) = 0

两边减去

1

1 + tan (2 x) - 1 = 0 - 1

化简

tan (2 x) = - 1

tan (2 x) = - 1

tan (2 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}

2 x = \frac{3 π}{4} + πn

2 x = \frac{3 π}{4} + πn

解

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

2 x = \frac{3 π}{4} + πn

两边除以

2

2 x = \frac{3 π}{4} + πn

两边除以

2

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

化简

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

\frac{\frac{3 π}{4}}{2} + \frac{πn}{2}

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

使用分式法则:

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

= \frac{3 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{3 π}{8}

= \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

合并所有解

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

作图

流行的例子

solvefor x,0.05cos(x)=-sin(x)

so l v e f or x, 0.05 cos (x) = - sin (x)

cos (θ) = \frac{3}{10}

tan (x) = \frac{8}{17}

sin(x-90)=cos(x)

sin (x - 9 0^{\circ}) = cos (x)

cos (x) = \frac{67}{77}