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受欢迎的三角函数 >

sin(4x)+cos(4x)=1

解答

sin (4 x) + cos (4 x) = 1

解答

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

+1

度数

x = 0^{\circ} + 9 0^{\circ} n, x = 22. 5^{\circ} + 9 0^{\circ} n

求解步骤

sin (4 x) + cos (4 x) = 1

使用三角恒等式改写

sin (4 x) + cos (4 x)

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

sin (4 x) + cos (4 x)

改写为

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

使用角和恒等式:

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

= 2 sin (4 x + \frac{π}{4})

= 2 sin (4 x + \frac{π}{4})

2 sin (4 x + \frac{π}{4}) = 1

两边除以

2

2 sin (4 x + \frac{π}{4}) = 1

两边除以

2

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

化简

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

化简

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

\frac{2 sin ( 4 x + \frac{π}{4} )}{2}

约分:

2

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

sin (4 x + \frac{π}{4}) = \frac{2}{2}

sin (4 x + \frac{π}{4}) = \frac{2}{2}

sin (4 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}

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

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

解

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

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

两边减去

\frac{π}{4}

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

化简

4 x = 2 πn

两边除以

4

4 x = 2 πn

两边除以

4

\frac{4 x}{4} = \frac{2 πn}{4}

化简

x = \frac{πn}{2}

x = \frac{πn}{2}

解

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

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

将

\frac{π}{4}

到右边

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

两边减去

\frac{π}{4}

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

化简

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

化简

4 x + \frac{π}{4} - \frac{π}{4} : 4 x

4 x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= 4 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}

4 x = 2 πn + \frac{π}{2}

4 x = 2 πn + \frac{π}{2}

4 x = 2 πn + \frac{π}{2}

两边除以

4

4 x = 2 πn + \frac{π}{2}

两边除以

4

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

化简

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

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} : \frac{πn}{2} + \frac{π}{8}

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

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

使用分式法则:

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

= \frac{π}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{π}{8}

= \frac{πn}{2} + \frac{π}{8}

x = \frac{πn}{2} + \frac{π}{8}

x = \frac{πn}{2} + \frac{π}{8}

x = \frac{πn}{2} + \frac{π}{8}

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

作图

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