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### Transcript of Assignment 6 - prl.res.in jayesh/ آ  Assignment 6 Group 5: Apurv & Sanjay Question 2 i. If sin(...

• 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

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