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受欢迎的三角函数 >

cos^2(3/4 pi+x)+sin^2(7/4 pi-x)=1

解答

cos^{2} (\frac{3}{4} π + x) + sin^{2} (\frac{7}{4} π - x) = 1

解答

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

+1

度数

x = 18 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (\frac{3}{4} π + x) + sin^{2} (\frac{7}{4} π - x) = 1

使用三角恒等式改写

cos^{2} (\frac{3}{4} π + x) + sin^{2} (\frac{7}{4} π - x) = 1

使用三角恒等式改写

cos (\frac{3}{4} π + x)

使用角和恒等式:

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

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

化简

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

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

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

cos (\frac{3}{4} π) cos (x)

乘

\frac{3}{4} π : \frac{3 π}{4}

\frac{3}{4} π

分式相乘:

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

= \frac{3 π}{4}

= cos (\frac{3 π}{4}) cos (x)

化简

cos (\frac{3 π}{4}) : - \frac{2}{2}

cos (\frac{3 π}{4})

使用以下普通恒等式:

cos (\frac{3 π}{4}) = - \frac{2}{2}

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{2}{2}

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

分式相乘:

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

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

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

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

sin (\frac{3}{4} π) sin (x)

乘

\frac{3}{4} π : \frac{3 π}{4}

\frac{3}{4} π

分式相乘:

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

= \frac{3 π}{4}

= sin (\frac{3 π}{4}) sin (x)

化简

sin (\frac{3 π}{4}) : \frac{2}{2}

sin (\frac{3 π}{4})

使用以下普通恒等式:

sin (\frac{3 π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

分式相乘:

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

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

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

使用法则

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

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

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

使用角差恒等式:

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

= sin (\frac{7}{4} π) cos (x) - cos (\frac{7}{4} π) sin (x)

化简

sin (\frac{7}{4} π) cos (x) - cos (\frac{7}{4} π) sin (x) : \frac{- 2 cos ( x ) - 2 sin ( x )}{2}

sin (\frac{7}{4} π) cos (x) - cos (\frac{7}{4} π) sin (x)

sin (\frac{7}{4} π) cos (x) = - \frac{2 cos ( x )}{2}

sin (\frac{7}{4} π) cos (x)

乘

\frac{7}{4} π : \frac{7 π}{4}

\frac{7}{4} π

分式相乘:

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

= \frac{7 π}{4}

= sin (\frac{7 π}{4}) cos (x)

sin (\frac{7 π}{4}) = - \frac{2}{2}

sin (\frac{7 π}{4})

使用三角恒等式改写:

sin (π) cos (\frac{3 π}{4}) + cos (π) sin (\frac{3 π}{4})

sin (\frac{7 π}{4})

将

sin (\frac{7 π}{4})

写为

sin (π + \frac{3 π}{4})

= sin (π + \frac{3 π}{4})

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{3 π}{4}) + cos (π) sin (\frac{3 π}{4})

= sin (π) cos (\frac{3 π}{4}) + cos (π) sin (\frac{3 π}{4})

使用以下普通恒等式:

sin (π) = 0

sin (π)

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}

= 0

使用以下普通恒等式:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

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{2}{2}

使用以下普通恒等式:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

使用以下普通恒等式:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

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}

= \frac{2}{2}

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

化简

= - \frac{2}{2}

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

分式相乘:

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

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

= - \frac{2 cos ( x )}{2} - cos (π \frac{7}{4}) sin (x)

cos (\frac{7}{4} π) sin (x) = \frac{2 sin ( x )}{2}

cos (\frac{7}{4} π) sin (x)

乘

\frac{7}{4} π : \frac{7 π}{4}

\frac{7}{4} π

分式相乘:

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

= \frac{7 π}{4}

= cos (\frac{7 π}{4}) sin (x)

cos (\frac{7 π}{4}) = \frac{2}{2}

cos (\frac{7 π}{4})

使用三角恒等式改写:

cos (π) cos (\frac{3 π}{4}) - sin (π) sin (\frac{3 π}{4})

cos (\frac{7 π}{4})

将

cos (\frac{7 π}{4})

写为

cos (π + \frac{3 π}{4})

= cos (π + \frac{3 π}{4})

使用角和恒等式:

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

= cos (π) cos (\frac{3 π}{4}) - sin (π) sin (\frac{3 π}{4})

= cos (π) cos (\frac{3 π}{4}) - sin (π) sin (\frac{3 π}{4})

使用以下普通恒等式:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

使用以下普通恒等式:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

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{2}{2}

使用以下普通恒等式:

sin (π) = 0

sin (π)

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}

= 0

使用以下普通恒等式:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

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}

= \frac{2}{2}

= (- 1) (- \frac{2}{2}) - 0 \cdot \frac{2}{2}

化简

= \frac{2}{2}

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

分式相乘:

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

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

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

使用法则

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

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

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

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

化简

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

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

同类项相加:

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

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

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

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

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

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

因式分解出通项

2

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

消掉

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

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

使用根式运算法则:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

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

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

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

数字相减:

1 - \frac{1}{2} = \frac{1}{2}

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

使用根式运算法则:

a^{\frac{1}{n}} = n a

2^{\frac{1}{2}} = 2

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

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

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

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

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

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

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

(2)^{2} : 2

使用根式运算法则:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 2

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

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

分式相乘:

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

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

约分:

2

= (cos (x) + sin (x))^{2}

(cos (x) + sin (x))^{2} = 1

(cos (x) + sin (x))^{2} = 1

两边减去

1

(cos (x) + sin (x))^{2} - 1 = 0

分解

(cos (x) + sin (x))^{2} - 1 : (cos (x) + sin (x) + 1) (cos (x) + sin (x) - 1)

(cos (x) + sin (x))^{2} - 1

将

1

改写为

1^{2}

= (cos (x) + sin (x))^{2} - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(cos (x) + sin (x))^{2} - 1^{2} = ((cos (x) + sin (x)) + 1) ((cos (x) + sin (x)) - 1)

= ((cos (x) + sin (x)) + 1) ((cos (x) + sin (x)) - 1)

整理后得

= (cos (x) + sin (x) + 1) (cos (x) + sin (x) - 1)

(cos (x) + sin (x) + 1) (cos (x) + sin (x) - 1) = 0

分别求解每个部分

cos (x) + sin (x) + 1 = 0 or cos (x) + sin (x) - 1 = 0

cos (x) + sin (x) + 1 = 0 : x = 2 πn + π, x = 2 πn + \frac{3 π}{2}

cos (x) + sin (x) + 1 = 0

使用三角恒等式改写

cos (x) + sin (x) + 1

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

sin (x) + cos (x)

改写为

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

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 (x) + sin (\frac{π}{4}) cos (x))

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (x + \frac{π}{4})

= 1 + 2 sin (x + \frac{π}{4})

1 + 2 sin (x + \frac{π}{4}) = 0

将

1

到右边

1 + 2 sin (x + \frac{π}{4}) = 0

两边减去

1

1 + 2 sin (x + \frac{π}{4}) - 1 = 0 - 1

化简

2 sin (x + \frac{π}{4}) = - 1

2 sin (x + \frac{π}{4}) = - 1

两边除以

2

2 sin (x + \frac{π}{4}) = - 1

两边除以

2

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

化简

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

化简

\frac{2 sin ( x + \frac{π}{4} )}{2} : sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

约分:

2

= sin (x + \frac{π}{4})

化简

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

\frac{- 1}{2}

使用分式法则:

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

= - \frac{1}{2}

- \frac{1}{2}

有理化:

- \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}

= - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \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}

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn, x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn, x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

解

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn : x = 2 πn + π

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn

将

\frac{π}{4}

到右边

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn

两边减去

\frac{π}{4}

x + \frac{π}{4} - \frac{π}{4} = \frac{5 π}{4} + 2 πn - \frac{π}{4}

化简

x + \frac{π}{4} - \frac{π}{4} = \frac{5 π}{4} + 2 πn - \frac{π}{4}

化简

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= x

化简

\frac{5 π}{4} + 2 πn - \frac{π}{4} : 2 πn + π

\frac{5 π}{4} + 2 πn - \frac{π}{4}

对同类项分组

= 2 πn - \frac{π}{4} + \frac{5 π}{4}

合并分式

- \frac{π}{4} + \frac{5 π}{4} : π

使用法则

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

= \frac{- π + 5 π}{4}

同类项相加:

- π + 5 π = 4 π

= \frac{4 π}{4}

数字相除:

\frac{4}{4} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

解

x + \frac{π}{4} = \frac{7 π}{4} + 2 πn : x = 2 πn + \frac{3 π}{2}

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

将

\frac{π}{4}

到右边

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

两边减去

\frac{π}{4}

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

化简

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

化简

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= x

化简

\frac{7 π}{4} + 2 πn - \frac{π}{4} : 2 πn + \frac{3 π}{2}

\frac{7 π}{4} + 2 πn - \frac{π}{4}

对同类项分组

= 2 πn - \frac{π}{4} + \frac{7 π}{4}

合并分式

- \frac{π}{4} + \frac{7 π}{4} : \frac{3 π}{2}

使用法则

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

= \frac{- π + 7 π}{4}

同类项相加:

- π + 7 π = 6 π

= \frac{6 π}{4}

约分:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

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

cos (x) + sin (x) - 1 = 0 : x = 2 πn, x = 2 πn + \frac{π}{2}

cos (x) + sin (x) - 1 = 0

使用三角恒等式改写

cos (x) + sin (x) - 1

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

sin (x) + cos (x)

改写为

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

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 (x) + sin (\frac{π}{4}) cos (x))

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (x + \frac{π}{4})

= - 1 + 2 sin (x + \frac{π}{4})

- 1 + 2 sin (x + \frac{π}{4}) = 0

将

1

到右边

- 1 + 2 sin (x + \frac{π}{4}) = 0

两边加上

1

- 1 + 2 sin (x + \frac{π}{4}) + 1 = 0 + 1

化简

2 sin (x + \frac{π}{4}) = 1

2 sin (x + \frac{π}{4}) = 1

两边除以

2

2 sin (x + \frac{π}{4}) = 1

两边除以

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{1}{2}

化简

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{1}{2}

化简

\frac{2 sin ( x + \frac{π}{4} )}{2} : sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

约分:

2

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

sin (x + \frac{π}{4}) = \frac{2}{2}

sin (x + \frac{π}{4}) = \frac{2}{2}

sin (x + \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}

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

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

解

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

x + \frac{π}{4} = \frac{π}{4} + 2 πn

两边减去

\frac{π}{4}

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

化简

x = 2 πn

解

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

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

将

\frac{π}{4}

到右边

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

两边减去

\frac{π}{4}

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

化简

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

化简

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= x

化简

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

\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} : \frac{π}{2}

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 = 2 π

= \frac{2 π}{4}

约分:

2

= \frac{π}{2}

= 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn, x = 2 πn + \frac{π}{2}

合并所有解

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

作图

流行的例子

sin(4x)=-sin(x)

sin (4 x) = - sin (x)

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

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

cos(x)(tan(x)-sqrt(3))=0

cos (x) (tan (x) - 3) = 0

2+2cos(t)=6cos(t)

2 + 2 cos (t) = 6 cos (t)

2cos(x)+1=cos(x)

2 cos (x) + 1 = cos (x)