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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

cos (2 x) + cos (x) + 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})

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

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

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

同类项相加:

2 x - x = x

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

同类项相加:

2 x + x = 3 x

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

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

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

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

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

同类项相加:

2 x + x = 3 x

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

同类项相加:

2 x - x = x

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

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

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

分解

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

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

因式分解出通项

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

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

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

分别求解每个部分

cos (\frac{3 x}{2}) = 0 or cos (\frac{x}{2}) + sin (\frac{x}{2}) = 0

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

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

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

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

解

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

\frac{3 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

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

化简

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

化简

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

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

数字相乘:

2 \cdot 3 = 6

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 x

化简

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分式相乘:

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

= \frac{π 2}{2}

约分:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

两边除以

3

3 x = π + 4 πn

两边除以

3

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

化简

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

解

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

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

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

化简

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

化简

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

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

数字相乘:

2 \cdot 3 = 6

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 x

化简

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

分式相乘:

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

= \frac{3 π 2}{2}

约分:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

两边除以

3

3 x = 3 π + 4 πn

两边除以

3

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

化简

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

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

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

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

使用三角恒等式改写

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

在两边除以

cos (\frac{x}{2}), cos (\frac{x}{2}) \neq = 0

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

化简

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

使用基本三角恒等式:

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

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

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

将

1

到右边

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

两边减去

1

1 + tan (\frac{x}{2}) - 1 = 0 - 1

化简

tan (\frac{x}{2}) = - 1

tan (\frac{x}{2}) = - 1

tan (\frac{x}{2}) = - 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}

\frac{x}{2} = \frac{3 π}{4} + πn

\frac{x}{2} = \frac{3 π}{4} + πn

解

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

\frac{x}{2} = \frac{3 π}{4} + πn

在两边乘以

2

\frac{x}{2} = \frac{3 π}{4} + πn

在两边乘以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

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

2 \cdot \frac{3 π}{4} + 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

分式相乘:

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

= \frac{3 π 2}{4}

数字相乘:

3 \cdot 2 = 6

= \frac{6 π}{4}

约分:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

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

作图

流行的例子

(cos(x)=1)=(sec(x)=1)

(cos (x) = 1) = (sec (x) = 1)

cot(x/3)=(-1)/(sqrt(3))

cot (\frac{x}{3}) = \frac{- 1}{3}

cos^2(θ)=sin^2(θ)+1,0<= θ<= 360

cos^{2} (θ) = sin^{2} (θ) + 1, 0^{\circ} \leq θ \leq 36 0^{\circ}

tan(2x)(1-tan^2(x))= 2/(sqrt(3))

tan (2 x) (1 - tan^{2} (x)) = \frac{2}{3}

tan (x) = 25762