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受欢迎的三角函数 >

sin(x)-cos(x)=0.5

解答

sin (x) - cos (x) = 0.5

解答

x = 0.36136 \dots + 2 πn + \frac{π}{4}, x = π - 0.36136 \dots + 2 πn + \frac{π}{4}

+1

度数

x = 65.70481 \dots^{\circ} + 36 0^{\circ} n, x = 204.29518 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (x) - cos (x) = 0.5

使用三角恒等式改写

sin (x) - cos (x)

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})

= 2 sin (x - \frac{π}{4})

2 sin (x - \frac{π}{4}) = 0.5

两边除以

2

2 sin (x - \frac{π}{4}) = 0.5

两边除以

2

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{0.5}{2}

化简

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{0.5}{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{0.5}{2} : 0.35355 \dots

\frac{0.5}{2}

转换为小数形式

2 = 1.41421 \dots

= \frac{0.5}{1.41421 \dots}

数字相除:

\frac{0.5}{1.41421 \dots} = 0.35355 \dots

= 0.35355 \dots

sin (x - \frac{π}{4}) = 0.35355 \dots

sin (x - \frac{π}{4}) = 0.35355 \dots

sin (x - \frac{π}{4}) = 0.35355 \dots

使用反三角函数性质

sin (x - \frac{π}{4}) = 0.35355 \dots

sin (x - \frac{π}{4}) = 0.35355 \dots

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x - \frac{π}{4} = arcsin (0.35355 \dots) + 2 πn, x - \frac{π}{4} = π - arcsin (0.35355 \dots) + 2 πn

x - \frac{π}{4} = arcsin (0.35355 \dots) + 2 πn, x - \frac{π}{4} = π - arcsin (0.35355 \dots) + 2 πn

解

x - \frac{π}{4} = arcsin (0.35355 \dots) + 2 πn : x = arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

x - \frac{π}{4} = arcsin (0.35355 \dots) + 2 πn

将

\frac{π}{4}

到右边

x - \frac{π}{4} = arcsin (0.35355 \dots) + 2 πn

两边加上

\frac{π}{4}

x - \frac{π}{4} + \frac{π}{4} = arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

化简

x = arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

x = arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

解

x - \frac{π}{4} = π - arcsin (0.35355 \dots) + 2 πn : x = π - arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

x - \frac{π}{4} = π - arcsin (0.35355 \dots) + 2 πn

将

\frac{π}{4}

到右边

x - \frac{π}{4} = π - arcsin (0.35355 \dots) + 2 πn

两边加上

\frac{π}{4}

x - \frac{π}{4} + \frac{π}{4} = π - arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

化简

x = π - arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

x = π - arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

x = arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}, x = π - arcsin (0.35355 \dots) + 2 πn + \frac{π}{4}

以小数形式表示解

x = 0.36136 \dots + 2 πn + \frac{π}{4}, x = π - 0.36136 \dots + 2 πn + \frac{π}{4}

作图

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