X2 T01 04 mod arg relations
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Transcript of X2 T01 04 mod arg relations
![Page 1: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/1.jpg)
Mod-Arg Relations
![Page 2: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/2.jpg)
Mod-Arg Relations 2121 1 zzzz
![Page 3: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/3.jpg)
Mod-Arg Relations 2121 1 zzzz
2121 argargarg zzzz
![Page 4: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/4.jpg)
Mod-Arg Relations 2121 1 zzzz
2121 argargarg zzzz
Proof:
![Page 5: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/5.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
Proof:
![Page 6: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/6.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
![Page 7: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/7.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
2121212121 sinsinsincoscossincoscos iirr
![Page 8: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/8.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
![Page 9: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/9.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
![Page 10: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/10.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
![Page 11: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/11.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
21
2121
argarg arg
zzzz
![Page 12: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/12.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
21
2121
argarg arg
zzzz
NOTE: it follows that;
![Page 13: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/13.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
21
2121
argarg arg
zzzz
NOTE: it follows that;nn zzzzzzzz 321321
![Page 14: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/14.jpg)
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz
21
2121
zzrrzz
Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
212121 sincos irr
21
2121
argarg arg
zzzz
NOTE: it follows that;nn zzzzzzzz 321321
nn zzzzzzzz argargargargarg 321321
![Page 15: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/15.jpg)
2
1
2
1 2zz
zz
![Page 16: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/16.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
![Page 17: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/17.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
Proof:
![Page 18: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/18.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
22
22
222
111
2
1
sincossincos
sincossincos
ii
irir
zz
Proof:
![Page 19: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/19.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
22
22
222
111
2
1
sincossincos
sincossincos
ii
irir
zz
Proof:
22
22
21212121
2
1
sincossinsinsincoscossincoscos
ii
rr
![Page 20: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/20.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
22
22
222
111
2
1
sincossincos
sincossincos
ii
irir
zz
Proof:
22
22
21212121
2
1
sincossinsinsincoscossincoscos
ii
rr
212121212
1 sincoscossinsinsincoscos irr
![Page 21: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/21.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
22
22
222
111
2
1
sincossincos
sincossincos
ii
irir
zz
Proof:
22
22
21212121
2
1
sincossinsinsincoscossincoscos
ii
rr
212121212
1 sincoscossinsinsincoscos irr
21212
1 sincos irr
![Page 22: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/22.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
22
22
222
111
2
1
sincossincos
sincossincos
ii
irir
zz
2
1
2
1
2
1
zzrr
zz
Proof:
22
22
21212121
2
1
sincossinsinsincoscossincoscos
ii
rr
212121212
1 sincoscossinsinsincoscos irr
21212
1 sincos irr
![Page 23: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/23.jpg)
2
1
2
1 2zz
zz
212
1 argargarg zzzz
22
22
222
111
2
1
sincossincos
sincossincos
ii
irir
zz
2
1
2
1
2
1
zzrr
zz
Proof:
22
22
21212121
2
1
sincossinsinsincoscossincoscos
ii
rr
212121212
1 sincoscossinsinsincoscos irr
21212
1 sincos irr
21
212
1
argarg
arg
zzzz
![Page 24: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/24.jpg)
NOTE: it follows that;
![Page 25: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/25.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
![Page 26: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/26.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
![Page 27: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/27.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
![Page 28: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/28.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
![Page 29: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/29.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
i
iize.g.2325 ofargument and modulus theFind
![Page 30: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/30.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
![Page 31: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/31.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
![Page 32: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/32.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
10
![Page 33: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/33.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
10
32tan
21tan
51tanarg 111z
![Page 34: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/34.jpg)
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
i
iize.g.2325 ofargument and modulus theFind
22
2222
231215
z
13526
10
32tan
21tan
51tanarg 111z
84175
1433621539111
![Page 35: X2 T01 04 mod arg relations](https://reader033.fdocument.pub/reader033/viewer/2022060116/557e78e2d8b42a03668b5172/html5/thumbnails/35.jpg)
Exercise 4B; evens
Exercise 4C; 1 to 10 evens, 12 to 15 all