1
00:00:00,000 --> 00:00:02,300
如果在街道上我走过来

2
00:00:02,300 --> 00:00:04,310
问你-

3
00:00:04,310 --> 00:00:06,640
我不想写得那么复杂

4
00:00:06,640 --> 00:00:11,750
sinπ/4等于多少

5
00:00:11,750 --> 00:00:14,800
很明显 我们讨论的是弧度

6
00:00:14,800 --> 00:00:16,810
要么记住结果

7
00:00:16,810 --> 00:00:19,210
要么画出单位圆

8
00:00:19,210 --> 00:00:21,420
画得不太标准

9
00:00:21,420 --> 00:00:22,390
你们明白意思就行

10
00:00:22,390 --> 00:00:26,080
π/4弧度

11
00:00:26,080 --> 00:00:29,070
也就是45°

12
00:00:29,070 --> 00:00:31,820
半径为1

13
00:00:31,820 --> 00:00:34,700
那么sin可以定义为

14
00:00:34,700 --> 00:00:36,350
单位圆上的纵坐标

15
00:00:36,350 --> 00:00:38,750
只需要知道这个值

16
00:00:38,750 --> 00:00:40,330
就可以了

17
00:00:40,330 --> 00:00:42,690
这个角是45°

18
00:00:42,690 --> 00:00:44,560
将三角形画得大一些

19
00:00:44,560 --> 00:00:47,080
三角形像这样

20
00:00:47,080 --> 00:00:48,970
这是45°

21
00:00:48,970 --> 00:00:51,010
这是45°

22
00:00:51,010 --> 00:00:53,830
这是90°

23
00:00:53,830 --> 00:00:57,350
可以求解45°45°90°三角形

24
00:00:57,350 --> 00:00:58,560
斜边长度为1

25
00:00:58,560 --> 00:01:00,740
这两个边设为x

26
00:01:00,740 --> 00:01:01,940
长度相等

27
00:01:01,940 --> 00:01:04,940
是等腰三角形

28
00:01:04,940 --> 00:01:06,590
底角相等

29
00:01:06,590 --> 00:01:07,820
那么

30
00:01:07,820 --> 00:01:11,420
x方+x方等于1的平方

31
00:01:11,420 --> 00:01:12,700
也就是等于1

32
00:01:12,700 --> 00:01:14,900
2倍的x方等于1

33
00:01:14,900 --> 00:01:17,160
x方等于1/2

34
00:01:17,160 --> 00:01:20,370
x等于1/2的平方根

35
00:01:20,370 --> 00:01:22,060
等于1/根2

36
00:01:22,060 --> 00:01:25,430
可以化为有理数

37
00:01:25,430 --> 00:01:31,390
乘以√2/√2

38
00:01:31,390 --> 00:01:34,470
得到x等于√2/2

39
00:01:34,470 --> 00:01:38,990
这里高是√2/2

40
00:01:38,990 --> 00:01:40,360
如果想知道这个长度

41
00:01:40,360 --> 00:01:41,550
也是√2/2

42
00:01:41,550 --> 00:01:43,100
我们只关心这个高度

43
00:01:43,100 --> 00:01:45,880
因为正弦值 sinπ/4

44
00:01:45,880 --> 00:01:47,970
就是这个高度

45
00:01:47,970 --> 00:01:49,050
纵坐标值

46
00:01:49,050 --> 00:01:52,830
等于√2/2

47
00:01:52,830 --> 00:01:54,030
这些都是复习

48
00:01:54,030 --> 00:02:00,090
在单位圆那一集视频中学过

49
00:02:00,090 --> 00:02:02,410
但是如果是其他人

50
00:02:02,410 --> 00:02:04,540
另一天我走过来

51
00:02:04,540 --> 00:02:12,250
向你请教

52
00:02:12,250 --> 00:02:14,590
arcsin√2/2等于多少

53
00:02:14,590 --> 00:02:16,350
arcsin是什么？

54
00:02:16,350 --> 00:02:17,050
你们会感到困惑

55
00:02:17,050 --> 00:02:19,090
你们可能会想 我知道一个角度的正弦值

56
00:02:19,090 --> 00:02:22,850
但这个是Sal定义的一个新的

57
00:02:22,850 --> 00:02:24,290
三角函数

58
00:02:24,290 --> 00:02:26,120
你们应该意识到

59
00:02:26,120 --> 00:02:28,350
当单词前缀是arc时

60
00:02:28,350 --> 00:02:30,890
有时指的是反正弦

61
00:02:30,890 --> 00:02:33,120
可以简写为

62
00:02:33,120 --> 00:02:36,130
sin-1√2/2

63
00:02:36,130 --> 00:02:37,930
等于多少？

64
00:02:37,930 --> 00:02:40,830
其实问题是

65
00:02:40,830 --> 00:02:43,870
正弦值是根2/2

66
00:02:43,870 --> 00:02:48,070
对应角度是多少

67
00:02:48,070 --> 00:02:51,530
也就是什么角度的

68
00:02:51,530 --> 00:02:54,580
正弦值是√2/2

69
00:02:54,580 --> 00:02:59,360
这些表述可以重写为

70
00:02:59,360 --> 00:03:01,570
我们来写一下

71
00:03:01,570 --> 00:03:04,890
这些表述可以重写为

72
00:03:04,890 --> 00:03:10,980
sin?=√2/2

73
00:03:10,980 --> 00:03:14,910
我认为这个问题

74
00:03:14,910 --> 00:03:16,010
更简单一些

75
00:03:16,010 --> 00:03:18,290
sin?=√2/2

76
00:03:18,290 --> 00:03:22,110
我说过sinπ/4等于

77
00:03:22,110 --> 00:03:23,400
√2/2

78
00:03:23,400 --> 00:03:25,140
在这种情况下

79
00:03:25,140 --> 00:03:28,220
我知道sinπ/4

80
00:03:28,220 --> 00:03:30,260
等于√2/2

81
00:03:30,260 --> 00:03:35,040
所以这个问号等于π/4

82
00:03:35,040 --> 00:03:39,450
也可以重写为

83
00:03:39,450 --> 00:03:47,180
arcsin√2/2等于

84
00:03:47,180 --> 00:03:51,560
π/4

85
00:03:51,560 --> 00:03:54,790
你们或许会说- 复习一下

86
00:03:54,790 --> 00:03:56,220
给定一个值

87
00:03:56,220 --> 00:03:58,090
我是指对于给定角度

88
00:03:58,090 --> 00:03:59,710
求其正弦值

89
00:03:59,710 --> 00:04:00,910
值是确定的

90
00:04:00,910 --> 00:04:02,840
但是你们倾向于-

91
00:04:02,840 --> 00:04:03,590
看一下

92
00:04:03,590 --> 00:04:05,060
从这里复习

93
00:04:05,060 --> 00:04:07,040
π/4是正确的（注意这里老师误说成π/2了）

94
00:04:07,040 --> 00:04:08,470
45°是正确的

95
00:04:08,470 --> 00:04:11,360
如果不断累加360°

96
00:04:11,360 --> 00:04:12,980
或者不断增加2π

97
00:04:12,980 --> 00:04:14,480
这些值都是正确的

98
00:04:14,480 --> 00:04:15,470
因为这些值对应

99
00:04:15,470 --> 00:04:18,760
单位圆上同一个点

100
00:04:18,760 --> 00:04:19,970
确实如此

101
00:04:19,970 --> 00:04:21,570
所有这些值

102
00:04:21,570 --> 00:04:24,720
都是正确的结果

103
00:04:24,720 --> 00:04:25,380
对吧？

104
00:04:25,380 --> 00:04:27,630
因为如果对任意这些角度求正弦值

105
00:04:27,630 --> 00:04:29,700
对不断累加360°后的角度

106
00:04:29,700 --> 00:04:31,500
求其正弦值

107
00:04:31,500 --> 00:04:33,240
都会得到√2/2

108
00:04:33,240 --> 00:04:34,550
这是个问题

109
00:04:34,550 --> 00:04:35,520
不应该存在这样的函数

110
00:04:35,520 --> 00:04:36,990
如果写成函数

111
00:04:36,990 --> 00:04:39,580
不应该存在函数f(x)

112
00:04:39,580 --> 00:04:42,350
同一x对应多个函数值

113
00:04:42,350 --> 00:04:44,570
x可以取π/4

114
00:04:44,570 --> 00:04:47,980
可以取π/4+2π

115
00:04:47,980 --> 00:04:52,320
也可以取π/4+4π

116
00:04:52,320 --> 00:04:54,720
为得到正确的函数

117
00:04:54,720 --> 00:04:57,840
为使反正弦函数有意义

118
00:04:57,840 --> 00:04:59,930
需要限定取值范围

119
00:04:59,930 --> 00:05:02,620
将取值范围

120
00:05:02,620 --> 00:05:04,240
限定为最普遍的情况

121
00:05:04,240 --> 00:05:06,240
现在限定取值范围

122
00:05:06,240 --> 00:05:08,720
换句话说

123
00:05:08,720 --> 00:05:10,360
定义域是什么？

124
00:05:10,360 --> 00:05:12,410
如果求arcsin

125
00:05:12,410 --> 00:05:17,320
对x求arcsin

126
00:05:17,320 --> 00:05:20,240
设arcsin等于θ

127
00:05:20,240 --> 00:05:22,030
定义域是什么？

128
00:05:22,030 --> 00:05:23,770
x的取值范围是什么？

129
00:05:23,770 --> 00:05:26,140
x可以取多少？

130
00:05:26,140 --> 00:05:30,010
对任意角度求正弦值

131
00:05:30,010 --> 00:05:33,380
得到的结果在1到-1之间

132
00:05:33,380 --> 00:05:36,560
所以x取值

133
00:05:36,560 --> 00:05:37,640
-1≤x

134
00:05:37,640 --> 00:05:39,130
x≤1

135
00:05:39,130 --> 00:05:40,530
这就是定义域

136
00:05:40,530 --> 00:05:43,910
为使函数存在

137
00:05:43,910 --> 00:05:45,190
需要限定取值范围

138
00:05:45,190 --> 00:05:46,520
可能的取值

139
00:05:46,520 --> 00:05:47,660
需要限定取值范围

140
00:05:47,660 --> 00:05:48,800
对于arcsin

141
00:05:48,800 --> 00:05:51,300
一般限定在第一和

142
00:05:51,300 --> 00:05:52,170
第四象限

143
00:05:52,170 --> 00:05:54,250
将取值角度限定在

144
00:05:54,250 --> 00:05:58,560
单位圆上这一区域内

145
00:05:58,560 --> 00:06:02,650
所以θ取值

146
00:06:02,650 --> 00:06:04,210
θ≤π/2

147
00:06:04,210 --> 00:06:10,960
-π/2≤θ

148
00:06:10,960 --> 00:06:14,260
由此可知arcsin是多少了

149
00:06:14,260 --> 00:06:16,890
再解一道题

150
00:06:16,890 --> 00:06:18,850
换个地方来写

151
00:06:18,850 --> 00:06:21,540
再求一个arcsin

152
00:06:21,540 --> 00:06:22,920
问题是

153
00:06:22,920 --> 00:06:36,100
arcsin(-√3/2)

154
00:06:36,100 --> 00:06:37,920
如果你记得

155
00:06:37,920 --> 00:06:39,830
那么马上可以得到sinx

156
00:06:39,830 --> 00:06:41,630
或者sinθ=-√3/2

157
00:06:41,630 --> 00:06:42,380
问题解决了

158
00:06:42,380 --> 00:06:43,840
但是我没记住结果

159
00:06:43,840 --> 00:06:46,980
我就需要画出单位圆

160
00:06:46,980 --> 00:06:48,210
帮助求解arcsin

161
00:06:48,210 --> 00:06:50,730
只需要画出单位圆的

162
00:06:50,730 --> 00:06:53,280
第一和第四象限

163
00:06:53,280 --> 00:06:54,830
这是y轴

164
00:06:54,830 --> 00:06:56,880
这是x轴

165
00:06:56,880 --> 00:06:59,050
x和y

166
00:06:59,050 --> 00:07:00,840
题目对应的点在哪里？

167
00:07:00,840 --> 00:07:02,960
某角度的正弦值

168
00:07:02,960 --> 00:07:04,590
等于-√3/2

169
00:07:04,590 --> 00:07:06,590
意味着单位圆上点的纵坐标

170
00:07:06,590 --> 00:07:09,400
等于-√3/2

171
00:07:09,400 --> 00:07:14,150
就是这个点

172
00:07:14,150 --> 00:07:18,840
纵坐标是-√3/2

173
00:07:18,840 --> 00:07:20,020
这是对应的点

174
00:07:20,020 --> 00:07:24,320
对应的角度是多少？

175
00:07:24,320 --> 00:07:25,570
想一下

176
00:07:25,670 --> 00:07:29,850
纵坐标是-√3/2

177
00:07:29,850 --> 00:07:32,990
就是这个角

178
00:07:32,990 --> 00:07:35,700
这个角是负的

179
00:07:35,700 --> 00:07:38,530
因为沿顺时针方向在x轴下方

180
00:07:38,530 --> 00:07:40,240
为求解

181
00:07:40,240 --> 00:07:43,440
画一个小三角形

182
00:07:43,440 --> 00:07:45,800
换种颜色

183
00:07:45,800 --> 00:07:46,740
这个三角形

184
00:07:46,740 --> 00:07:53,050
用蓝色画

185
00:07:53,050 --> 00:07:54,560
将三角形放大

186
00:07:54,560 --> 00:07:56,420
像这样

187
00:07:56,420 --> 00:07:57,250
这是θ

188
00:07:57,250 --> 00:07:58,730
这是θ

189
00:07:58,730 --> 00:08:00,320
这个边长是多少？

190
00:08:00,320 --> 00:08:02,850
也就是纵坐标

191
00:08:02,850 --> 00:08:03,950
可以称为

192
00:08:03,950 --> 00:08:06,160
是√3/2

193
00:08:06,160 --> 00:08:07,730
其实是负值 因为是在x轴下面

194
00:08:07,730 --> 00:08:09,140
但只是为表征这个角

195
00:08:09,140 --> 00:08:11,670
我们知道这个角是负的

196
00:08:11,670 --> 00:08:14,020
当看到√3/2时

197
00:08:14,020 --> 00:08:16,870
希望你们会想到这是30°60°90°三角形

198
00:08:16,870 --> 00:08:18,080
这个边是√3/2

199
00:08:18,080 --> 00:08:20,150
这个边是1/2

200
00:08:20,150 --> 00:08:21,420
当然 这个边是1

201
00:08:21,420 --> 00:08:23,220
因为单位圆

202
00:08:23,220 --> 00:08:24,360
半径为1

203
00:08:24,360 --> 00:08:26,380
30°60°90°三角形

204
00:08:26,380 --> 00:08:28,400
对边是√3/2

205
00:08:28,400 --> 00:08:30,430
θ=60°

206
00:08:30,430 --> 00:08:32,270
这个角是30°

207
00:08:32,270 --> 00:08:35,310
我们得到θ=60°

208
00:08:35,310 --> 00:08:36,220
这是其大小

209
00:08:36,220 --> 00:08:37,660
因为在x轴下面

210
00:08:37,660 --> 00:08:39,150
所以是-60°

211
00:08:39,150 --> 00:08:43,120
所以θ=-60°

212
00:08:43,120 --> 00:08:44,570
如果化为弧度

213
00:08:44,570 --> 00:08:45,400
还不够好

214
00:08:45,400 --> 00:08:48,610
可以乘以100-

215
00:08:48,610 --> 00:08:54,590
抱歉 乘以π弧度比上180°

216
00:08:54,590 --> 00:08:55,820
消去角度

217
00:08:55,820 --> 00:08:59,620
得到θ=-π/3

218
00:08:59,620 --> 00:09:03,900
弧度

219
00:09:03,900 --> 00:09:05,270
那么现在

220
00:09:05,270 --> 00:09:09,100
可以得出结论

221
00:09:09,100 --> 00:09:15,960
arcsin(-√3/2)

222
00:09:15,960 --> 00:09:19,590
等于-π/3弧度

223
00:09:19,590 --> 00:09:24,980
或者写成sin-1(-√3/2)等于

224
00:09:24,980 --> 00:09:30,830
-π/3弧度

225
00:09:30,830 --> 00:09:33,270
为验证这个结果

226
00:09:33,270 --> 00:09:35,390
我们用计算器算一下

227
00:09:35,390 --> 00:09:38,290
设为弧度模式

228
00:09:38,290 --> 00:09:39,420
检查一下

229
00:09:39,420 --> 00:09:40,940
点击Per second mode键

230
00:09:40,940 --> 00:09:42,550
弧度模式下

231
00:09:42,550 --> 00:09:45,490
希望得到正确的结果

232
00:09:45,490 --> 00:09:47,510
计算反正弦

233
00:09:47,510 --> 00:09:51,130
sin-1

234
00:09:51,130 --> 00:09:58,650
（-√3/2）

235
00:09:58,650 --> 00:10:03,880
等于-1.04

236
00:10:03,880 --> 00:10:05,080
可知

237
00:10:05,080 --> 00:10:10,720
等于-1.04rad

238
00:10:10,720 --> 00:10:14,020
那么π/3应该等于1.04

239
00:10:14,020 --> 00:10:16,130
验算一下

240
00:10:16,130 --> 00:10:23,520
-π/3

241
00:10:23,520 --> 00:10:24,950
等于多少？

242
00:10:24,950 --> 00:10:26,810
结果相同

243
00:10:26,810 --> 00:10:28,580
计算器给出相同的结果

244
00:10:28,580 --> 00:10:30,140
但是计算器有缺点

245
00:10:30,140 --> 00:10:31,920
因为计算器不能显示出

246
00:10:31,920 --> 00:10:34,120
这就是-π/3