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Mod-Arg Relations
Mod-Arg Relations 2121 1 zzzz
Mod-Arg Relations 2121 1 zzzz
2121 argargarg zzzz
Mod-Arg Relations 2121 1 zzzz
2121 argargarg zzzz
Proof:
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
Proof:
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
2121212121 sinsinsincoscossincoscos iirr
Mod-Arg Relations 2121 1 zzzz
222111 andlet cisrzcisrz
2121 argargarg zzzz
22211121 sincossincos irirzz Proof:
2121212121 sinsinsincoscossincoscos iirr 2121212121 sincoscossinsinsincoscos irr
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
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
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
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;
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
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
2
1
2
1 2zz
zz
2
1
2
1 2zz
zz
212
1 argargarg zzzz
2
1
2
1 2zz
zz
212
1 argargarg zzzz
Proof:
2
1
2
1 2zz
zz
212
1 argargarg zzzz
22
22
222
111
2
1
sincossincos
sincossincos
ii
irir
zz
Proof:
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
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
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
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
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
NOTE: it follows that;
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
NOTE: it follows that;
43
21
43
21
zzzz
zzzz
432143
21 argargargargarg zzzzzzzz
nn zz 3
znzn argarg
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
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
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
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
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
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
Exercise 4B; evens
Exercise 4C; 1 to 10 evens, 12 to 15 all