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15.8*sin(30x-90)+5=17

解答

15.8 \cdot sin (30 x - 90) + 5 = 17

解答

x = 3 + \frac{πn}{15} + \frac{0.86253 \dots}{30}, x = \frac{π}{30} + \frac{πn}{15} - \frac{0.86253 \dots}{30} + 3

+1

度数

x = 173.53465 \dots^{\circ} + 1 2^{\circ} n, x = 176.24001 \dots^{\circ} + 1 2^{\circ} n

求解步骤

15.8 sin (30 x - 90) + 5 = 17

在两边乘以

10

15.8 sin (30 x - 90) + 5 = 17

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

15.8 sin (30 x - 90) \cdot 10 + 5 \cdot 10 = 17 \cdot 10

整理后得

158 sin (30 x - 90) + 50 = 170

158 sin (30 x - 90) + 50 = 170

将

50

到右边

158 sin (30 x - 90) + 50 = 170

两边减去

50

158 sin (30 x - 90) + 50 - 50 = 170 - 50

化简

158 sin (30 x - 90) = 120

158 sin (30 x - 90) = 120

两边除以

158

158 sin (30 x - 90) = 120

两边除以

158

\frac{158 sin ( 30 x - 90 )}{158} = \frac{120}{158}

化简

sin (30 x - 90) = \frac{60}{79}

sin (30 x - 90) = \frac{60}{79}

使用反三角函数性质

sin (30 x - 90) = \frac{60}{79}

sin (30 x - 90) = \frac{60}{79}

的通解

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

30 x - 90 = arcsin (\frac{60}{79}) + 2 πn, 30 x - 90 = π - arcsin (\frac{60}{79}) + 2 πn

30 x - 90 = arcsin (\frac{60}{79}) + 2 πn, 30 x - 90 = π - arcsin (\frac{60}{79}) + 2 πn

解

30 x - 90 = arcsin (\frac{60}{79}) + 2 πn : x = 3 + \frac{πn}{15} + \frac{arcsin ( \frac{60}{79} )}{30}

30 x - 90 = arcsin (\frac{60}{79}) + 2 πn

将

90

到右边

30 x - 90 = arcsin (\frac{60}{79}) + 2 πn

两边加上

90

30 x - 90 + 90 = arcsin (\frac{60}{79}) + 2 πn + 90

化简

30 x = arcsin (\frac{60}{79}) + 2 πn + 90

30 x = arcsin (\frac{60}{79}) + 2 πn + 90

两边除以

30

30 x = arcsin (\frac{60}{79}) + 2 πn + 90

两边除以

30

\frac{30 x}{30} = \frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30}

化简

\frac{30 x}{30} = \frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30}

化简

\frac{30 x}{30} : x

\frac{30 x}{30}

数字相除:

\frac{30}{30} = 1

= x

化简

\frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30} : 3 + \frac{πn}{15} + \frac{arcsin ( \frac{60}{79} )}{30}

\frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30}

对同类项分组

= \frac{90}{30} + \frac{2 πn}{30} + \frac{arcsin ( \frac{60}{79} )}{30}

数字相除:

\frac{90}{30} = 3

= 3 + \frac{2 πn}{30} + \frac{arcsin ( \frac{60}{79} )}{30}

消掉

\frac{2 πn}{30} : \frac{πn}{15}

\frac{2 πn}{30}

约分:

2

= \frac{πn}{15}

= 3 + \frac{πn}{15} + \frac{arcsin ( \frac{60}{79} )}{30}

x = 3 + \frac{πn}{15} + \frac{arcsin ( \frac{60}{79} )}{30}

x = 3 + \frac{πn}{15} + \frac{arcsin ( \frac{60}{79} )}{30}

x = 3 + \frac{πn}{15} + \frac{arcsin ( \frac{60}{79} )}{30}

解

30 x - 90 = π - arcsin (\frac{60}{79}) + 2 πn : x = \frac{π}{30} + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30} + 3

30 x - 90 = π - arcsin (\frac{60}{79}) + 2 πn

将

90

到右边

30 x - 90 = π - arcsin (\frac{60}{79}) + 2 πn

两边加上

90

30 x - 90 + 90 = π - arcsin (\frac{60}{79}) + 2 πn + 90

化简

30 x = π - arcsin (\frac{60}{79}) + 2 πn + 90

30 x = π - arcsin (\frac{60}{79}) + 2 πn + 90

两边除以

30

30 x = π - arcsin (\frac{60}{79}) + 2 πn + 90

两边除以

30

\frac{30 x}{30} = \frac{π}{30} - \frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30}

化简

\frac{30 x}{30} = \frac{π}{30} - \frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30}

化简

\frac{30 x}{30} : x

\frac{30 x}{30}

数字相除:

\frac{30}{30} = 1

= x

化简

\frac{π}{30} - \frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30} : \frac{π}{30} + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30} + 3

\frac{π}{30} - \frac{arcsin ( \frac{60}{79} )}{30} + \frac{2 πn}{30} + \frac{90}{30}

对同类项分组

= \frac{π}{30} + \frac{90}{30} + \frac{2 πn}{30} - \frac{arcsin ( \frac{60}{79} )}{30}

数字相除:

\frac{90}{30} = 3

= \frac{π}{30} + 3 + \frac{2 πn}{30} - \frac{arcsin ( \frac{60}{79} )}{30}

消掉

\frac{2 πn}{30} : \frac{πn}{15}

\frac{2 πn}{30}

约分:

2

= \frac{πn}{15}

= \frac{π}{30} + 3 + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30}

对同类项分组

= \frac{π}{30} + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30} + 3

x = \frac{π}{30} + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30} + 3

x = \frac{π}{30} + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30} + 3

x = \frac{π}{30} + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30} + 3

x = 3 + \frac{πn}{15} + \frac{arcsin ( \frac{60}{79} )}{30}, x = \frac{π}{30} + \frac{πn}{15} - \frac{arcsin ( \frac{60}{79} )}{30} + 3

以小数形式表示解

x = 3 + \frac{πn}{15} + \frac{0.86253 \dots}{30}, x = \frac{π}{30} + \frac{πn}{15} - \frac{0.86253 \dots}{30} + 3

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