查看“︁三角函數”︁的源代码

[[Wikibooks:首页|首页]] > [[Subject:索引|維基書架]] > [[数学书架]] > [[三角函数]]
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{{Wikipedia|三角函数}}

== 三角函數的基本定義 ==

[[Image:Z-Triangle.PNG|thumb|圖一]]

三角函數最一開始是用來表示角度和直角三角形三邊邊長關係的式子，直角三角形中的<math>\sin x</math> 和<math>\cos x </math>可由畢氏定理給出它的定義：
若一個直角三角形，它的一個銳角角度為<math>x</math>，此角的對邊為<math>a</math>，鄰邊為<math>b</math>，斜邊為<math>c</math>（如圖所示），則：

<math>\sin x = \frac {a}{c}=\frac{\text{对 边 长 }}{\text{斜 边 长 }}</math>

<math>\cos x = \frac {b}{c}=\frac {\text{鄰 边 长 }}{\text{斜 边 长 }}</math>

<math>\tan x = \frac {a}{b}=\frac {\text{对 边 长 }}{\text{鄰 边 长 }}</math>

因此得到正弦函數<math>\sin x</math>和餘弦函數<math>\cos x</math>的定義.

當<math>270^\circ < \phi \le 360^\circ</math>時，<math>-1 < \sin \phi \le 0</math>且<math>0 < \cos \phi \le 1</math>

==弧度制與角度制的轉換==
一个角度制数值所对应的弧度制数值等于单位圆中圆心角角度与该角度制数值相同时该圆心角所对应的弧长。用<math>{rad}</math>表示弧度制数值，用<math>{deg}</math>表示角度制数值，二者转换关系为：

:<math>{rad}=\pi{\frac{deg}{180^{\circ}}}</math>
常用的弧度转换公式：

:<math>2\pi \leftrightarrow 360^{\circ}</math>

:<math>\pi \leftrightarrow 180^{\circ}</math>

:<math>\frac{\pi}{2} \leftrightarrow 90^{\circ}</math>

:<math>\frac{\pi}{3} \leftrightarrow 60^{\circ}</math>

:<math>\frac{\pi}{4} \leftrightarrow 45^{\circ}</math>

:<math>\frac{\pi}{6} \leftrightarrow 30^{\circ}</math>

== 主要的公式 ==
=== 倒数关系 ===
<math>\sin\alpha=\frac{1}{\csc\alpha}</math>

<math>\cos\beta=\frac{1}{\sec\beta}</math>

<math>\tan\theta=\frac{1}{\cot\theta}</math>

=== 平方相加 ===
<math>\cos^2 \theta + \sin^2 \theta= 1</math><br>
<math>\tan^2 \theta + 1= \sec^2 \theta</math><br>
<math>1 + \cot^2 \theta= \csc^2 \theta</math>

=== 和角公式 ===
<math>\sin(\alpha\pm\beta)=\sin\alpha \cos\beta\pm\cos\alpha \sin\beta</math>

<math>\cos(\alpha\pm\beta)=\cos\alpha \cos\beta\mp\sin\alpha \sin\beta</math>

<math>\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha \tan\beta}</math>

=== 倍角公式 & 半角公式 ===

'''2倍角公式''':

<math>\sin2\theta = 2\sin\theta \cos\theta</math>

<math>\cos2\theta = \cos^2 \theta - \sin^2 \theta=2\cos^2 \theta-1=1-2\sin^2 \theta</math>

<math>\tan2\theta = \frac{2\tan\theta}{1- \tan^2 \theta}</math>

'''3倍角公式''':

<math>\cos3\theta=4\cos^3\theta -3\cos\theta</math>

<math>\sin3\theta=-4\sin^3 \theta +3\sin\theta</math>

'''半角公式''':

<math>\sin\frac{\theta}{2}=\pm\sqrt\frac{1-\cos\theta}{2}</math>

<math>\cos\frac{\theta}{2}=\pm\sqrt\frac{1+\cos\theta}{2}</math>

<math>\tan\frac{\theta}{2}=\pm\sqrt\frac{1-\cos\theta}{1+cos\theta}=\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}</math>

'''積化和差''':

<math>\sin\alpha\cos\beta=\frac{\sin(\alpha+\beta)+\sin(\alpha-\beta)}{2}</math>

<math>\cos\alpha\sin\beta=\frac{\sin(\alpha+\beta)-\sin(\alpha-\beta)}{2}</math>

<math>\cos\alpha\cos\beta=\frac{\cos(\alpha+\beta)+\cos(\alpha-\beta)}{2}</math>

<math>\sin\alpha\sin\beta=\frac{\cos(\alpha-\beta)-\cos(\alpha+\beta)}{2}</math>

'''和差化積''':

<math>\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}</math>

<math>\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}</math>

<math>\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}</math>

<math>\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}</math>

=== 其他公式 ===

万能公式：

<math>\sin2\alpha=\frac{2\tan\alpha}{1+\tan^2 \alpha}</math>

<math>\cos2\alpha=\frac{1-\tan^2 \alpha}{1+\tan^2 \alpha}</math>

<math>\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2 \alpha}</math>

平方差公式：

<math>\sin(\alpha+\beta)\cdot \sin(\alpha-\beta)=\sin^2 \alpha-\sin^2 \beta</math>

<math>\cos(\alpha+\beta)\cdot \cos(\alpha-\beta)=\cos^2 \alpha-\sin^2 \beta</math>

降次升角公式：

<math>\cos^2 \alpha=\frac{1+\cos2\alpha}{2}</math>

<math>\sin^2 \alpha=\frac{1-\cos2\alpha}{2}</math>

辅助角公式：

<math>a \sin\alpha + b \cos\alpha=\sqrt{a^2 +b^2}\cdot\sin\left(\alpha+\arctan\frac{b}{a}\right)</math>

== 三角函數的一些進階定理 ==

=== 正弦定理： ===
若為任意三角形的三邊<math>a</math>,<math>b</math>,<math>c</math>，<math>A</math>,<math>B</math>,<math>C</math>分別為<math>a</math>,<math>b</math>,<math>c</math>的對角，<math>R</math>為此三角形的外接圓半徑，則存有以下關係式：
<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} =2R=\frac{a+b+c}{\sin A+\sin B+\sin C}</math>

由勾股定理得
<math>\sin^2\theta+\cos^2\theta=1</math>
当<math>\theta</math>的终边与坐标轴重合时，这个公式也成立。

根据三角函数的定义，当<math>\theta\neq k\pi +\frac{\pi}{2}</math>（<math>k\not\in\mathbb{Z}</math>）时，有
<math>\frac{\sin\theta}{\cos\theta}=\tan\theta</math>

这就是说，同一个角<math>\theta</math>的正弦、余弦的平方和等于1，商等于角<math>\theta</math>的正切。

== 三角分析 ==

面积公式：

<math>S=\frac{1}{2}ah=\frac{1}{2}bc\sin A=\frac{1}{2}ab\sin C=\frac{1}{2}ac\sin B=\frac{abc}{4R}=2R^2 \sin A\sin B\sin C</math>

三角形形状的判定:

1) <math>c</math>为三角形<math>ABC</math>最大边时

<math>a^2 +b^2 =c^2</math>三角形<math>ABC</math>为直角三角形

<math>a^2 +b^2 >c^2</math>三角形<math>ABC</math>为锐角三角形

<math>a^2 +b^2 <c^2</math>三角形<math>ABC</math>为钝角三角形

2) <math>A<B</math>推出 <math>\sin A<\sin B</math>

== 三角函數應用 ==
在实际生活中的应用很广，举个例子吧：
1.测量建筑物的高度，用到仰角的正弦值；
2.高架桥的两端的引桥，用到的坡度，是三角函数的正切值；
3.汽车的侧方停车也要看角度后进对吗？
4.公路的拐弯处也要科学运用角度的大小问题
=== 三角測量 ===

=== 其他應用 ===

= 附錄表 =

==附錄─三角函數表==

{| class="wikitable"
|度
|分
|sin
|cos
|tan
|cot
|sec
|csc
|度
|分
|-
|0
|00
|0.0000
|1.0000
|0.0000
|未定義
|1.0000
|未定義
|90
|00
|-
|0
|10
|0.0029
|1.0000
|0.0029
|343.7737
|1.0000
|343.7752
|89
|50
|-
|0
|20
|0.0058
|1.0000
|0.0058
|171.8854
|1.0000
|171.8883
|89
|40
|-
|0
|30
|0.0087
|1.0000
|0.0087
|114.5887
|1.0000
|114.5930
|89
|30
|-
|0
|40
|0.0116
|0.9999
|0.0116
|85.9398
|1.0001
|85.9456
|89
|20
|-
|0
|50
|0.0145
|0.9999
|0.0145
|68.7501
|1.0001
|68.7574
|89
|10
|-
|1
|00
|0.0175
|0.9998
|0.0175
|57.2900
|1.0002
|57.2987
|89
|00
|-
|1
|10
|0.0204
|0.9998
|0.0204
|49.1039
|1.0002
|49.1141
|88
|50
|-
|1
|20
|0.0233
|0.9997
|0.0233
|42.9641
|1.0003
|42.9757
|88
|40
|-
|1
|30
|0.0262
|0.9997
|0.0262
|38.1885
|1.0003
|38.2016
|88
|30
|-
|1
|40
|0.0291
|0.9996
|0.0291
|34.3678
|1.0004
|34.3823
|88
|20
|-
|1
|50
|0.0320
|0.9995
|0.0320
|31.2416
|1.0005
|31.2576
|88
|10
|-
|2
|00
|0.0349
|0.9994
|0.0349
|28.6363
|1.0006
|28.6537
|88
|00
|-
|2
|10
|0.0378
|0.9993
|0.0378
|26.4316
|1.0007
|26.4505
|87
|50
|-
|2
|20
|0.0407
|0.9992
|0.0407
|24.5418
|1.0008
|24.5621
|87
|40
|-
|2
|30
|0.0436
|0.9990
|0.0437
|22.9038
|1.0010
|22.9256
|87
|30
|-
|2
|40
|0.0465
|0.9989
|0.0466
|21.4704
|1.0011
|21.4937
|87
|20
|-
|2
|50
|0.0494
|0.9988
|0.0495
|20.2056
|1.0012
|20.2303
|87
|10
|-
|3
|00
|0.0523
|0.9986
|0.0524
|19.0811
|1.0014
|19.1073
|87
|00
|-
|3
|10
|0.0552
|0.9985
|0.0553
|18.0750
|1.0015
|18.1026
|86
|50
|-
|3
|20
|0.0581
|0.9983
|0.0582
|17.1693
|1.0017
|17.1984
|86
|40
|-
|3
|30
|0.0610
|0.9981
|0.0612
|16.3499
|1.0019
|16.3804
|86
|30
|-
|3
|40
|0.0640
|0.9980
|0.0641
|15.6048
|1.0021
|15.6368
|86
|20
|-
|3
|50
|0.0669
|0.9978
|0.0670
|14.9244
|1.0022
|14.9579
|86
|10
|-
|4
|00
|0.0698
|0.9976
|0.0699
|14.3007
|1.0024
|14.3356
|86
|00
|-
|4
|10
|0.0727
|0.9974
|0.0729
|13.7267
|1.0027
|13.7631
|85
|50
|-
|4
|20
|0.0756
|0.9971
|0.0758
|13.1969
|1.0029
|13.2347
|85
|40
|-
|4
|30
|0.0785
|0.9969
|0.0787
|12.7062
|1.0031
|12.7455
|85
|30
|-
|4
|40
|0.0814
|0.9967
|0.0816
|12.2505
|1.0033
|12.2913
|85
|20
|-
|4
|50
|0.0843
|0.9964
|0.0846
|11.8262
|1.0036
|11.8684
|85
|10
|-
|5
|00
|0.0872
|0.9962
|0.0875
|11.4301
|1.0038
|11.4737
|85
|00
|-
|5
|10
|0.0901
|0.9959
|0.0904
|11.0594
|1.0041
|11.1045
|84
|50
|-
|5
|20
|0.0929
|0.9957
|0.0934
|10.7119
|1.0043
|10.7585
|84
|40
|-
|5
|30
|0.0958
|0.9954
|0.0963
|10.3854
|1.0046
|10.4334
|84
|30
|-
|5
|40
|0.0987
|0.9951
|0.0992
|10.0780
|1.0049
|10.1275
|84
|20
|-
|5
|50
|0.1016
|0.9948
|0.1022
|9.7882
|1.0052
|9.8391
|84
|10
|-
|6
|00
|0.1045
|0.9945
|0.1051
|9.5144
|1.0055
|9.5668
|84
|00
|-
|6
|10
|0.1074
|0.9942
|0.1080
|9.2553
|1.0058
|9.3092
|83
|50
|-
|6
|20
|0.1103
|0.9939
|0.1110
|9.0098
|1.0061
|9.0652
|83
|40
|-
|6
|30
|0.1132
|0.9936
|0.1139
|8.7769
|1.0065
|8.8337
|83
|30
|-
|6
|40
|0.1161
|0.9932
|0.1169
|8.5555
|1.0068
|8.6138
|83
|20
|-
|6
|50
|0.1190
|0.9929
|0.1198
|8.3450
|1.0072
|8.4047
|83
|10
|-
|7
|00
|0.1219
|0.9925
|0.1228
|8.1443
|1.0075
|8.2055
|83
|00
|-
|7
|10
|0.1248
|0.9922
|0.1257
|7.9530
|1.0079
|8.0156
|82
|50
|-
|7
|20
|0.1276
|0.9918
|0.1287
|7.7704
|1.0082
|7.8344
|82
|40
|-
|7
|30
|0.1305
|0.9914
|0.1317
|7.5958
|1.0086
|7.6613
|82
|30
|-
|7
|40
|0.1334
|0.9911
|0.1346
|7.4287
|1.0090
|7.4957
|82
|20
|-
|7
|50
|0.1363
|0.9907
|0.1376
|7.2687
|1.0094
|7.3372
|82
|10
|-
|8
|00
|0.1392
|0.9903
|0.1405
|7.1154
|1.0098
|7.1853
|82
|00
|-
|8
|10
|0.1421
|0.9899
|0.1435
|6.9682
|1.0102
|7.0396
|81
|50
|-
|8
|20
|0.1449
|0.9894
|0.1465
|6.8269
|1.0107
|6.8998
|81
|40
|-
|8
|30
|0.1478
|0.9890
|0.1495
|6.6912
|1.0111
|6.7655
|81
|30
|-
|8
|40
|0.1507
|0.9886
|0.1524
|6.5606
|1.0116
|6.6363
|81
|20
|-
|8
|50
|0.1536
|0.9881
|0.1554
|6.4348
|1.0120
|6.5121
|81
|10
|-
|9
|00
|0.1564
|0.9877
|0.1584
|6.3138
|1.0125
|6.3925
|81
|00
|-
|9
|10
|0.1593
|0.9872
|0.1614
|6.1970
|1.0129
|6.2772
|80
|50
|-
|9
|20
|0.1622
|0.9868
|0.1644
|6.0844
|1.0134
|6.1661
|80
|40
|-
|9
|30
|0.1650
|0.9863
|0.1673
|5.9758
|1.0139
|6.0589
|80
|30
|-
|9
|40
|0.1679
|0.9858
|0.1703
|5.8708
|1.0144
|5.9554
|80
|20
|-
|9
|50
|0.1708
|0.9853
|0.1733
|5.7694
|1.0149
|5.8554
|80
|10
|-
|10
|00
|0.1736
|0.9848
|0.1763
|5.6713
|1.0154
|5.7588
|80
|00
|-
|10
|10
|0.1765
|0.9843
|0.1793
|5.5764
|1.0160
|5.6653
|79
|50
|-
|10
|20
|0.1794
|0.9838
|0.1823
|5.4845
|1.0165
|5.5749
|79
|40
|-
|10
|30
|0.1822
|0.9833
|0.1853
|5.3955
|1.0170
|5.4874
|79
|30
|-
|10
|40
|0.1851
|0.9827
|0.1883
|5.3093
|1.0176
|5.4026
|79
|20
|-
|10
|50
|0.1880
|0.9822
|0.1914
|5.2257
|1.0181
|5.3205
|79
|10
|-
|11
|00
|0.1908
|0.9816
|0.1944
|5.1446
|1.0187
|5.2408
|79
|00
|-
|11
|10
|0.1937
|0.9811
|0.1974
|5.0658
|1.0193
|5.1636
|78
|50
|-
|11
|20
|0.1965
|0.9805
|0.2004
|4.9894
|1.0199
|5.0886
|78
|40
|-
|11
|30
|0.1994
|0.9799
|0.2035
|4.9152
|1.0205
|5.0159
|78
|30
|-
|11
|40
|0.2022
|0.9793
|0.2065
|4.8430
|1.0211
|4.9452
|78
|20
|-
|11
|50
|0.2051
|0.9787
|0.2095
|4.7729
|1.0217
|4.8765
|78
|10
|-
|12
|00
|0.2079
|0.9781
|0.2126
|4.7046
|1.0223
|4.8097
|78
|00
|-
|12
|10
|0.2108
|0.9775
|0.2156
|4.6382
|1.0230
|4.7448
|77
|50
|-
|12
|20
|0.2136
|0.9769
|0.2186
|4.5736
|1.0236
|4.6817
|77
|40
|-
|12
|30
|0.2164
|0.9763
|0.2217
|4.5107
|1.0243
|4.6202
|77
|30
|-
|12
|40
|0.2193
|0.9757
|0.2247
|4.4494
|1.0249
|4.5604
|77
|20
|-
|12
|50
|0.2221
|0.9750
|0.2278
|4.3897
|1.0256
|4.5022
|77
|10
|-
|13
|00
|0.2250
|0.9744
|0.2309
|4.3315
|1.0263
|4.4454
|77
|00
|-
|13
|10
|0.2278
|0.9737
|0.2339
|4.2747
|1.0270
|4.3901
|76
|50
|-
|13
|20
|0.2306
|0.9730
|0.2370
|4.2193
|1.0277
|4.3362
|76
|40
|-
|13
|30
|0.2334
|0.9724
|0.2401
|4.1653
|1.0284
|4.2837
|76
|30
|-
|13
|40
|0.2363
|0.9717
|0.2432
|4.1126
|1.0291
|4.2324
|76
|20
|-
|13
|50
|0.2391
|0.9710
|0.2462
|4.0611
|1.0299
|4.1824
|76
|10
|-
|14
|00
|0.2419
|0.9703
|0.2493
|4.0108
|1.0306
|4.1336
|76
|00
|-
|14
|10
|0.2447
|0.9696
|0.2524
|3.9617
|1.0314
|4.0859
|75
|50
|-
|14
|20
|0.2476
|0.9689
|0.2555
|3.9136
|1.0321
|4.0394
|75
|40
|-
|14
|30
|0.2504
|0.9681
|0.2586
|3.8667
|1.0329
|3.9939
|75
|30
|-
|14
|40
|0.2532
|0.9674
|0.2617
|3.8208
|1.0337
|3.9495
|75
|20
|-
|14
|50
|0.2560
|0.9667
|0.2648
|3.7760
|1.0345
|3.9061
|75
|10
|-
|15
|00
|0.2588
|0.9659
|0.2679
|3.7321
|1.0353
|3.8637
|75
|00
|-
|15
|10
|0.2616
|0.9652
|0.2711
|3.6891
|1.0361
|3.8222
|74
|50
|-
|15
|20
|0.2644
|0.9644
|0.2742
|3.6470
|1.0369
|3.7817
|74
|40
|-
|15
|30
|0.2672
|0.9636
|0.2773
|3.6059
|1.0377
|3.7420
|74
|30
|-
|15
|40
|0.2700
|0.9628
|0.2805
|3.5656
|1.0386
|3.7032
|74
|20
|-
|15
|50
|0.2728
|0.9621
|0.2836
|3.5261
|1.0394
|3.6652
|74
|10
|-
|16
|00
|0.2756
|0.9613
|0.2867
|3.4874
|1.0403
|3.6280
|74
|00
|-
|16
|10
|0.2784
|0.9605
|0.2899
|3.4495
|1.0412
|3.5915
|73
|50
|-
|16
|20
|0.2812
|0.9596
|0.2931
|3.4124
|1.0421
|3.5559
|73
|40
|-
|16
|30
|0.2840
|0.9588
|0.2962
|3.3759
|1.0429
|3.5209
|73
|30
|-
|16
|40
|0.2868
|0.9580
|0.2994
|3.3402
|1.0439
|3.4867
|73
|20
|-
|16
|50
|0.2896
|0.9572
|0.3026
|3.3052
|1.0448
|3.4532
|73
|10
|-
|17
|00
|0.2924
|0.9563
|0.3057
|3.2709
|1.0457
|3.4203
|73
|00
|-
|17
|10
|0.2952
|0.9555
|0.3089
|3.2371
|1.0466
|3.3881
|72
|50
|-
|17
|20
|0.2979
|0.9546
|0.3121
|3.2041
|1.0476
|3.3565
|72
|40
|-
|17
|30
|0.3007
|0.9537
|0.3153
|3.1716
|1.0485
|3.3255
|72
|30
|-
|17
|40
|0.3035
|0.9528
|0.3185
|3.1397
|1.0495
|3.2951
|72
|20
|-
|17
|50
|0.3062
|0.9520
|0.3217
|3.1084
|1.0505
|3.2653
|72
|10
|-
|18
|00
|0.3090
|0.9511
|0.3249
|3.0777
|1.0515
|3.2361
|72
|00
|-
|18
|10
|0.3118
|0.9502
|0.3281
|3.0475
|1.0525
|3.2074
|71
|50
|-
|18
|20
|0.3145
|0.9492
|0.3314
|3.0178
|1.0535
|3.1792
|71
|40
|-
|18
|30
|0.3173
|0.9483
|0.3346
|2.9887
|1.0545
|3.1515
|71
|30
|-
|18
|40
|0.3201
|0.9474
|0.3378
|2.9600
|1.0555
|3.1244
|71
|20
|-
|18
|50
|0.3228
|0.9465
|0.3411
|2.9319
|1.0566
|3.0977
|71
|10
|-
|19
|00
|0.3256
|0.9455
|0.3443
|2.9042
|1.0576
|3.0716
|71
|00
|-
|19
|10
|0.3283
|0.9446
|0.3476
|2.8770
|1.0587
|3.0458
|70
|50
|-
|19
|20
|0.3311
|0.9436
|0.3508
|2.8502
|1.0598
|3.0206
|70
|40
|-
|19
|30
|0.3338
|0.9426
|0.3541
|2.8239
|1.0608
|2.9957
|70
|30
|-
|19
|40
|0.3365
|0.9417
|0.3574
|2.7980
|1.0619
|2.9713
|70
|20
|-
|19
|50
|0.3393
|0.9407
|0.3607
|2.7725
|1.0631
|2.9474
|70
|10
|-
|20
|00
|0.3420
|0.9397
|0.3640
|2.7475
|1.0642
|2.9238
|70
|00
|-
|20
|10
|0.3448
|0.9387
|0.3673
|2.7228
|1.0653
|2.9006
|69
|50
|-
|20
|20
|0.3475
|0.9377
|0.3706
|2.6985
|1.0665
|2.8779
|69
|40
|-
|20
|30
|0.3502
|0.9367
|0.3739
|2.6746
|1.0676
|2.8555
|69
|30
|-
|20
|40
|0.3529
|0.9356
|0.3772
|2.6511
|1.0688
|2.8334
|69
|20
|-
|20
|50
|0.3557
|0.9346
|0.3805
|2.6279
|1.0700
|2.8117
|69
|10
|-
|21
|00
|0.3584
|0.9336
|0.3839
|2.6051
|1.0711
|2.7904
|69
|00
|-
|21
|10
|0.3611
|0.9325
|0.3872
|2.5826
|1.0723
|2.7695
|68
|50
|-
|21
|20
|0.3638
|0.9315
|0.3906
|2.5605
|1.0736
|2.7488
|68
|40
|-
|21
|30
|0.3665
|0.9304
|0.3939
|2.5386
|1.0748
|2.7285
|68
|30
|-
|21
|40
|0.3692
|0.9293
|0.3973
|2.5172
|1.0760
|2.7085
|68
|20
|-
|21
|50
|0.3719
|0.9283
|0.4006
|2.4960
|1.0773
|2.6888
|68
|10
|-
|22
|00
|0.3746
|0.9272
|0.4040
|2.4751
|1.0785
|2.6695
|68
|00
|-
|22
|10
|0.3773
|0.9261
|0.4074
|2.4545
|1.0798
|2.6504
|67
|50
|-
|22
|20
|0.3800
|0.9250
|0.4108
|2.4342
|1.0811
|2.6316
|67
|40
|-
|22
|30
|0.3827
|0.9239
|0.4142
|2.4142
|1.0824
|2.6131
|67
|30
|-
|22
|40
|0.3854
|0.9228
|0.4176
|2.3945
|1.0837
|2.5949
|67
|20
|-
|22
|50
|0.3881
|0.9216
|0.4210
|2.3750
|1.0850
|2.5770
|67
|10
|-
|23
|00
|0.3907
|0.9205
|0.4245
|2.3559
|1.0864
|2.5593
|67
|00
|-
|23
|10
|0.3934
|0.9194
|0.4279
|2.3369
|1.0877
|2.5419
|66
|50
|-
|23
|20
|0.3961
|0.9182
|0.4314
|2.3183
|1.0891
|2.5247
|66
|40
|-
|23
|30
|0.3987
|0.9171
|0.4348
|2.2998
|1.0904
|2.5078
|66
|30
|-
|23
|40
|0.4014
|0.9159
|0.4383
|2.2817
|1.0918
|2.4912
|66
|20
|-
|23
|50
|0.4041
|0.9147
|0.4417
|2.2637
|1.0932
|2.4748
|66
|10
|-
|24
|00
|0.4067
|0.9135
|0.4452
|2.2460
|1.0946
|2.4586
|66
|00
|-
|24
|10
|0.4094
|0.9124
|0.4487
|2.2286
|1.0961
|2.4426
|65
|50
|-
|24
|20
|0.4120
|0.9112
|0.4522
|2.2113
|1.0975
|2.4269
|65
|40
|-
|24
|30
|0.4147
|0.9100
|0.4557
|2.1943
|1.0989
|2.4114
|65
|30
|-
|24
|40
|0.4173
|0.9088
|0.4592
|2.1775
|1.1004
|2.3961
|65
|20
|-
|24
|50
|0.4200
|0.9075
|0.4628
|2.1609
|1.1019
|2.3811
|65
|10
|-
|25
|00
|0.4226
|0.9063
|0.4663
|2.1445
|1.1034
|2.3662
|65
|00
|-
|25
|10
|0.4253
|0.9051
|0.4699
|2.1283
|1.1049
|2.3515
|64
|50
|-
|25
|20
|0.4279
|0.9038
|0.4734
|2.1123
|1.1064
|2.3371
|64
|40
|-
|25
|30
|0.4305
|0.9026
|0.4770
|2.0965
|1.1079
|2.3228
|64
|30
|-
|25
|40
|0.4331
|0.9013
|0.4806
|2.0809
|1.1095
|2.3088
|64
|20
|-
|25
|50
|0.4358
|0.9001
|0.4841
|2.0655
|1.1110
|2.2949
|64
|10
|-
|26
|00
|0.4384
|0.8988
|0.4877
|2.0503
|1.1126
|2.2812
|64
|00
|-
|26
|10
|0.4410
|0.8975
|0.4913
|2.0353
|1.1142
|2.2677
|63
|50
|-
|26
|20
|0.4436
|0.8962
|0.4950
|2.0204
|1.1158
|2.2543
|63
|40
|-
|26
|30
|0.4462
|0.8949
|0.4986
|2.0057
|1.1174
|2.2412
|63
|30
|-
|26
|40
|0.4488
|0.8936
|0.5022
|1.9912
|1.1190
|2.2282
|63
|20
|-
|26
|50
|0.4514
|0.8923
|0.5059
|1.9768
|1.1207
|2.2153
|63
|10
|-
|27
|00
|0.4540
|0.8910
|0.5095
|1.9626
|1.1223
|2.2027
|63
|00
|-
|27
|10
|0.4566
|0.8897
|0.5132
|1.9486
|1.1240
|2.1902
|62
|50
|-
|27
|20
|0.4592
|0.8884
|0.5169
|1.9347
|1.1257
|2.1779
|62
|40
|-
|27
|30
|0.4617
|0.8870
|0.5206
|1.9210
|1.1274
|2.1657
|62
|30
|-
|27
|40
|0.4643
|0.8857
|0.5243
|1.9074
|1.1291
|2.1537
|62
|20
|-
|27
|50
|0.4669
|0.8843
|0.5280
|1.8940
|1.1308
|2.1418
|62
|10
|-
|28
|00
|0.4695
|0.8829
|0.5317
|1.8807
|1.1326
|2.1301
|62
|00
|-
|28
|10
|0.4720
|0.8816
|0.5354
|1.8676
|1.1343
|2.1185
|61
|50
|-
|28
|20
|0.4746
|0.8802
|0.5392
|1.8546
|1.1361
|2.1070
|61
|40
|-
|28
|30
|0.4772
|0.8788
|0.5430
|1.8418
|1.1379
|2.0957
|61
|30
|-
|28
|40
|0.4797
|0.8774
|0.5467
|1.8291
|1.1397
|2.0846
|61
|20
|-
|28
|50
|0.4823
|0.8760
|0.5505
|1.8165
|1.1415
|2.0736
|61
|10
|-
|29
|00
|0.4848
|0.8746
|0.5543
|1.8040
|1.1434
|2.0627
|61
|00
|-
|29
|10
|0.4874
|0.8732
|0.5581
|1.7917
|1.1452
|2.0519
|60
|50
|-
|29
|20
|0.4899
|0.8718
|0.5619
|1.7796
|1.1471
|2.0413
|60
|40
|-
|29
|30
|0.4924
|0.8704
|0.5658
|1.7675
|1.1490
|2.0308
|60
|30
|-
|29
|40
|0.4950
|0.8689
|0.5696
|1.7556
|1.1509
|2.0204
|60
|20
|-
|29
|50
|0.4975
|0.8675
|0.5735
|1.7437
|1.1528
|2.0101
|60
|10
|-
|30
|00
|0.5000
|0.8660
|0.5774
|1.7321
|1.1547
|2.0000
|60
|00
|-
|30
|10
|0.5025
|0.8646
|0.5812
|1.7205
|1.1566
|1.9900
|59
|50
|-
|30
|20
|0.5050
|0.8631
|0.5851
|1.7090
|1.1586
|1.9801
|59
|40
|-
|30
|30
|0.5075
|0.8616
|0.5890
|1.6977
|1.1606
|1.9703
|59
|30
|-
|30
|40
|0.5100
|0.8601
|0.5930
|1.6864
|1.1626
|1.9606
|59
|20
|-
|30
|50
|0.5125
|0.8587
|0.5969
|1.6753
|1.1646
|1.9511
|59
|10
|-
|31
|00
|0.5150
|0.8572
|0.6009
|1.6643
|1.1666
|1.9416
|59
|00
|-
|31
|10
|0.5175
|0.8557
|0.6048
|1.6534
|1.1687
|1.9323
|58
|50
|-
|31
|20
|0.5200
|0.8542
|0.6088
|1.6426
|1.1707
|1.9230
|58
|40
|-
|31
|30
|0.5225
|0.8526
|0.6128
|1.6319
|1.1728
|1.9139
|58
|30
|-
|31
|40
|0.5250
|0.8511
|0.6168
|1.6212
|1.1749
|1.9048
|58
|20
|-
|31
|50
|0.5275
|0.8496
|0.6208
|1.6107
|1.1770
|1.8959
|58
|10
|-
|32
|00
|0.5299
|0.8480
|0.6249
|1.6003
|1.1792
|1.8871
|58
|00
|-
|32
|10
|0.5324
|0.8465
|0.6289
|1.5900
|1.1813
|1.8783
|57
|50
|-
|32
|20
|0.5348
|0.8450
|0.6330
|1.5798
|1.1835
|1.8697
|57
|40
|-
|32
|30
|0.5373
|0.8434
|0.6371
|1.5697
|1.1857
|1.8612
|57
|30
|-
|32
|40
|0.5398
|0.8418
|0.6412
|1.5597
|1.1879
|1.8527
|57
|20
|-
|32
|50
|0.5422
|0.8403
|0.6453
|1.5497
|1.1901
|1.8443
|57
|10
|-
|33
|00
|0.5446
|0.8387
|0.6494
|1.5399
|1.1924
|1.8361
|57
|00
|-
|33
|10
|0.5471
|0.8371
|0.6536
|1.5301
|1.1946
|1.8279
|56
|50
|-
|33
|20
|0.5495
|0.8355
|0.6577
|1.5204
|1.1969
|1.8198
|56
|40
|-
|33
|30
|0.5519
|0.8339
|0.6619
|1.5108
|1.1992
|1.8118
|56
|30
|-
|33
|40
|0.5544
|0.8323
|0.6661
|1.5013
|1.2015
|1.8039
|56
|20
|-
|33
|50
|0.5568
|0.8307
|0.6703
|1.4919
|1.2039
|1.7960
|56
|10
|-
|34
|00
|0.5592
|0.8290
|0.6745
|1.4826
|1.2062
|1.7883
|56
|00
|-
|34
|10
|0.5616
|0.8274
|0.6787
|1.4733
|1.2086
|1.7806
|55
|50
|-
|34
|20
|0.5640
|0.8258
|0.6830
|1.4641
|1.2110
|1.7730
|55
|40
|-
|34
|30
|0.5664
|0.8241
|0.6873
|1.4550
|1.2134
|1.7655
|55
|30
|-
|34
|40
|0.5688
|0.8225
|0.6916
|1.4460
|1.2158
|1.7581
|55
|20
|-
|34
|50
|0.5712
|0.8208
|0.6959
|1.4370
|1.2183
|1.7507
|55
|10
|-
|35
|00
|0.5736
|0.8192
|0.7002
|1.4281
|1.2208
|1.7434
|55
|00
|-
|35
|10
|0.5760
|0.8175
|0.7046
|1.4193
|1.2233
|1.7362
|54
|50
|-
|35
|20
|0.5783
|0.8158
|0.7089
|1.4106
|1.2258
|1.7291
|54
|40
|-
|35
|30
|0.5807
|0.8141
|0.7133
|1.4019
|1.2283
|1.7221
|54
|30
|-
|35
|40
|0.5831
|0.8124
|0.7177
|1.3934
|1.2309
|1.7151
|54
|20
|-
|35
|50
|0.5854
|0.8107
|0.7221
|1.3848
|1.2335
|1.7081
|54
|10
|-
|36
|00
|0.5878
|0.8090
|0.7265
|1.3764
|1.2361
|1.7013
|54
|00
|-
|36
|10
|0.5901
|0.8073
|0.7310
|1.3680
|1.2387
|1.6945
|53
|50
|-
|36
|20
|0.5925
|0.8056
|0.7355
|1.3597
|1.2413
|1.6878
|53
|40
|-
|36
|30
|0.5948
|0.8039
|0.7400
|1.3514
|1.2440
|1.6812
|53
|30
|-
|36
|40
|0.5972
|0.8021
|0.7445
|1.3432
|1.2467
|1.6746
|53
|20
|-
|36
|50
|0.5995
|0.8004
|0.7490
|1.3351
|1.2494
|1.6681
|53
|10
|-
|37
|00
|0.6018
|0.7986
|0.7536
|1.3270
|1.2521
|1.6616
|53
|00
|-
|37
|10
|0.6041
|0.7969
|0.7581
|1.3190
|1.2549
|1.6553
|52
|50
|-
|37
|20
|0.6065
|0.7951
|0.7627
|1.3111
|1.2577
|1.6489
|52
|40
|-
|37
|30
|0.6088
|0.7934
|0.7673
|1.3032
|1.2605
|1.6427
|52
|30
|-
|37
|40
|0.6111
|0.7916
|0.7720
|1.2954
|1.2633
|1.6365
|52
|20
|-
|37
|50
|0.6134
|0.7898
|0.7766
|1.2876
|1.2661
|1.6303
|52
|10
|-
|38
|00
|0.6157
|0.7880
|0.7813
|1.2799
|1.2690
|1.6243
|52
|00
|-
|38
|10
|0.6180
|0.7862
|0.7860
|1.2723
|1.2719
|1.6183
|51
|50
|-
|38
|20
|0.6202
|0.7844
|0.7907
|1.2647
|1.2748
|1.6123
|51
|40
|-
|38
|30
|0.6225
|0.7826
|0.7954
|1.2572
|1.2778
|1.6064
|51
|30
|-
|38
|40
|0.6248
|0.7808
|0.8002
|1.2497
|1.2807
|1.6005
|51
|20
|-
|38
|50
|0.6271
|0.7790
|0.8050
|1.2423
|1.2837
|1.5948
|51
|10
|-
|39
|00
|0.6293
|0.7771
|0.8098
|1.2349
|1.2868
|1.5890
|51
|00
|-
|39
|10
|0.6316
|0.7753
|0.8146
|1.2276
|1.2898
|1.5833
|50
|50
|-
|39
|20
|0.6338
|0.7735
|0.8195
|1.2203
|1.2929
|1.5777
|50
|40
|-
|39
|30
|0.6361
|0.7716
|0.8243
|1.2131
|1.2960
|1.5721
|50
|30
|-
|39
|40
|0.6383
|0.7698
|0.8292
|1.2059
|1.2991
|1.5666
|50
|20
|-
|39
|50
|0.6406
|0.7679
|0.8342
|1.1988
|1.3022
|1.5611
|50
|10
|-
|40
|00
|0.6428
|0.7660
|0.8391
|1.1918
|1.3054
|1.5557
|50
|00
|-
|40
|10
|0.6450
|0.7642
|0.8441
|1.1847
|1.3086
|1.5504
|49
|50
|-
|40
|20
|0.6472
|0.7623
|0.8491
|1.1778
|1.3118
|1.5450
|49
|40
|-
|40
|30
|0.6494
|0.7604
|0.8541
|1.1708
|1.3151
|1.5398
|49
|30
|-
|40
|40
|0.6517
|0.7585
|0.8591
|1.1640
|1.3184
|1.5345
|49
|20
|-
|40
|50
|0.6539
|0.7566
|0.8642
|1.1571
|1.3217
|1.5294
|49
|10
|-
|41
|00
|0.6561
|0.7547
|0.8693
|1.1504
|1.3250
|1.5243
|49
|00
|-
|41
|10
|0.6583
|0.7528
|0.8744
|1.1436
|1.3284
|1.5192
|48
|50
|-
|41
|20
|0.6604
|0.7509
|0.8796
|1.1369
|1.3318
|1.5141
|48
|40
|-
|41
|30
|0.6626
|0.7490
|0.8847
|1.1303
|1.3352
|1.5092
|48
|30
|-
|41
|40
|0.6648
|0.7470
|0.8899
|1.1237
|1.3386
|1.5042
|48
|20
|-
|41
|50
|0.6670
|0.7451
|0.8952
|1.1171
|1.3421
|1.4993
|48
|10
|-
|42
|00
|0.6691
|0.7431
|0.9004
|1.1106
|1.3456
|1.4945
|48
|00
|-
|42
|10
|0.6713
|0.7412
|0.9057
|1.1041
|1.3492
|1.4897
|47
|50
|-
|42
|20
|0.6734
|0.7392
|0.9110
|1.0977
|1.3527
|1.4849
|47
|40
|-
|42
|30
|0.6756
|0.7373
|0.9163
|1.0913
|1.3563
|1.4802
|47
|30
|-
|42
|40
|0.6777
|0.7353
|0.9217
|1.0850
|1.3600
|1.4755
|47
|20
|-
|42
|50
|0.6799
|0.7333
|0.9271
|1.0786
|1.3636
|1.4709
|47
|10
|-
|43
|00
|0.6820
|0.7314
|0.9325
|1.0724
|1.3673
|1.4663
|47
|00
|-
|43
|10
|0.6841
|0.7294
|0.9380
|1.0661
|1.3711
|1.4617
|46
|50
|-
|43
|20
|0.6862
|0.7274
|0.9435
|1.0599
|1.3748
|1.4572
|46
|40
|-
|43
|30
|0.6884
|0.7254
|0.9490
|1.0538
|1.3786
|1.4527
|46
|30
|-
|43
|40
|0.6905
|0.7234
|0.9545
|1.0477
|1.3824
|1.4483
|46
|20
|-
|43
|50
|0.6926
|0.7214
|0.9601
|1.0416
|1.3863
|1.4439
|46
|10
|-
|44
|00
|0.6947
|0.7193
|0.9657
|1.0355
|1.3902
|1.4396
|46
|00
|-
|44
|10
|0.6967
|0.7173
|0.9713
|1.0295
|1.3941
|1.4352
|45
|50
|-
|44
|20
|0.6988
|0.7153
|0.9770
|1.0235
|1.3980
|1.4310
|45
|40
|-
|44
|30
|0.7009
|0.7133
|0.9827
|1.0176
|1.4020
|1.4267
|45
|30
|-
|44
|40
|0.7030
|0.7112
|0.9884
|1.0117
|1.4061
|1.4225
|45
|20
|-
|44
|50
|0.7050
|0.7092
|0.9942
|1.0058
|1.4101
|1.4183
|45
|10
|-
|45
|00
|0.7071
|0.7071
|1.0000
|1.0000
|1.4142
|1.4142
|45
|00
|-
|度
|分
|cos
|sin
|cot
|tan
|csc
|sec
|度
|分
|}