Assignment 6

Group 5: Apurv & Sanjay

Question

2 i. If sin(θ + iφ) = ρ(cosα + i sinα), prove that ρ2 = 1/2[cosh 2φ− cos 2θ]

Solution

ρ(cosα + i sinα) = sin (θ + iφ) (0.0.1)

= sin θ cos iφ+ cos θ sin iφ (0.0.2)

= sin θ coshφ+ i cos θ sinhφ (0.0.3)

For two complex numbers to be equal one of the necessary condition is that their magni-

tude must be equal. Then,

ρ2(cos2 α + sin2 α) = sin2 θ cosh2 φ+ cos2 θ sinh2 φ (0.0.4)

ρ2 = (1− cos2 θ) cosh2 φ+ cos2 θ(1− cosh2 φ) (0.0.5)

= cosh2 φ− cos2 θ (0.0.6)

= 1 + cosh 2φ

2 − 1 + cos 2θ

2 (0.0.7)

= 1

2 [cosh 2φ− cos 2θ] Q.E.D (0.0.8)

1

ASSIGNMENT-6

Group-10(Bivin Geo George and Ritwik Mondal)

Question 7:

If

, then prove that

Answer:

Now,

Using the equation,

And similarly using

Therefore from previous,

Then take the conjugate of that,

Multiplying the two last equations we got that

That’s what it was in the question.

P1_G2.pdf P2_i_G5.pdf P2_ii_G4.pdf P3_i_G3.pdf P3_ii_G6_REMARK.pdf P4_i_G7.pdf P4_ii_G11.pdf P5_G8.pdf P6_G9.pdf P7_G10.pdf P8_G1.pdf