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受欢迎的三角函数 >

sin(2θ)-cos(2θ)=1

解答

sin (2 θ) - cos (2 θ) = 1

解答

θ = \frac{π}{4} + πn, θ = πn + \frac{π}{2}

+1

度数

θ = 4 5^{\circ} + 18 0^{\circ} n, θ = 9 0^{\circ} + 18 0^{\circ} n

求解步骤

sin (2 θ) - cos (2 θ) = 1

使用三角恒等式改写

sin (2 θ) - cos (2 θ)

sin (2 θ) - cos (2 θ) = 2 sin (2 θ - \frac{π}{4})

sin (2 θ) - cos (2 θ)

改写为

= 2 (\frac{1}{2} sin (2 θ) - \frac{1}{2} cos (2 θ))

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{1}{2}

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (2 θ) - sin (\frac{π}{4}) cos (2 θ))

使用角和恒等式:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (2 θ - \frac{π}{4})

= 2 sin (2 θ - \frac{π}{4})

2 sin (2 θ - \frac{π}{4}) = 1

两边除以

2

2 sin (2 θ - \frac{π}{4}) = 1

两边除以

2

\frac{2 sin ( 2 θ - \frac{π}{4} )}{2} = \frac{1}{2}

化简

\frac{2 sin ( 2 θ - \frac{π}{4} )}{2} = \frac{1}{2}

化简

\frac{2 sin ( 2 θ - \frac{π}{4} )}{2} : sin (2 θ - \frac{π}{4})

\frac{2 sin ( 2 θ - \frac{π}{4} )}{2}

约分:

2

= sin (2 θ - \frac{π}{4})

化简

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

乘以共轭根式

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (2 θ - \frac{π}{4}) = \frac{2}{2}

sin (2 θ - \frac{π}{4}) = \frac{2}{2}

sin (2 θ - \frac{π}{4}) = \frac{2}{2}

sin (2 θ - \frac{π}{4}) = \frac{2}{2}

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}

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

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

解

2 θ - \frac{π}{4} = \frac{π}{4} + 2 πn : θ = \frac{π}{4} + πn

2 θ - \frac{π}{4} = \frac{π}{4} + 2 πn

将

\frac{π}{4}

到右边

2 θ - \frac{π}{4} = \frac{π}{4} + 2 πn

两边加上

\frac{π}{4}

2 θ - \frac{π}{4} + \frac{π}{4} = \frac{π}{4} + 2 πn + \frac{π}{4}

化简

2 θ - \frac{π}{4} + \frac{π}{4} = \frac{π}{4} + 2 πn + \frac{π}{4}

化简

2 θ - \frac{π}{4} + \frac{π}{4} : 2 θ

2 θ - \frac{π}{4} + \frac{π}{4}

同类项相加:

- \frac{π}{4} + \frac{π}{4} = 0

= 2 θ

化简

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

\frac{π}{4} + 2 πn + \frac{π}{4}

对同类项分组

= \frac{π}{4} + \frac{π}{4} + 2 πn

合并分式

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

使用法则

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

= \frac{π + π}{4}

同类项相加:

π + π = 2 π

= \frac{2 π}{4}

约分:

2

= \frac{π}{2}

= \frac{π}{2} + 2 πn

2 θ = \frac{π}{2} + 2 πn

2 θ = \frac{π}{2} + 2 πn

2 θ = \frac{π}{2} + 2 πn

两边除以

2

2 θ = \frac{π}{2} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

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

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

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

使用分式法则:

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

= \frac{π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

解

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

2 θ - \frac{π}{4} = \frac{3 π}{4} + 2 πn

将

\frac{π}{4}

到右边

2 θ - \frac{π}{4} = \frac{3 π}{4} + 2 πn

两边加上

\frac{π}{4}

2 θ - \frac{π}{4} + \frac{π}{4} = \frac{3 π}{4} + 2 πn + \frac{π}{4}

化简

2 θ - \frac{π}{4} + \frac{π}{4} = \frac{3 π}{4} + 2 πn + \frac{π}{4}

化简

2 θ - \frac{π}{4} + \frac{π}{4} : 2 θ

2 θ - \frac{π}{4} + \frac{π}{4}

同类项相加:

- \frac{π}{4} + \frac{π}{4} = 0

= 2 θ

化简

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

\frac{3 π}{4} + 2 πn + \frac{π}{4}

对同类项分组

= 2 πn + \frac{π}{4} + \frac{3 π}{4}

合并分式

\frac{π}{4} + \frac{3 π}{4} : π

使用法则

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

= \frac{π + 3 π}{4}

同类项相加:

π + 3 π = 4 π

= \frac{4 π}{4}

数字相除:

\frac{4}{4} = 1

= π

= 2 πn + π

2 θ = 2 πn + π

2 θ = 2 πn + π

2 θ = 2 πn + π

两边除以

2

2 θ = 2 πn + π

两边除以

2

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

化简

θ = πn + \frac{π}{2}

θ = πn + \frac{π}{2}

θ = \frac{π}{4} + πn, θ = πn + \frac{π}{2}

作图

流行的例子

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cos (θ) = \frac{1}{3}

tan(x)+1/(tan(x))=2

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

4 csc (θ) - 5 = 3

csc(5x)=-1,0<= x<= 2pi

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-4*cos^2(x)+3=0

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