X2 t01 04 mod arg relations (2012)

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