Expert Answer to "Find modulus and argument of ω=sin⁡(P+Q)+i(1−cos⁡(P+Q))(cos⁡P+cos⁡Q)+i(sin⁡P−sin⁡Q)"

Tessa Leach

Answered question

2022-01-25

Find modulus and argument of

ω = \frac{\sin (P + Q) + i (1 - \cos (P + Q))}{(\cos P + \cos Q) + i (\sin P - \sin Q)}

coolbananas03ok

Beginner2022-01-26Added 20 answers

We have

N = \sin^{2} (P + Q) + {(1 - \cos (P + Q))}^{2} = \sin^{2} (P + Q) + \cos^{2} (P + Q) + 1 - 2 \cos (P + Q)

= 2 (1 - \cos (P + Q)) = 2 \cdot 2 \sin^{2} \frac{P + Q}{2} = 4 \sin^{2} \frac{P + Q}{2}

and

D = \cos^{2} P + \cos^{2} Q + \sin^{2} P + \sin^{2} Q + 2 (\cos P \cos Q - \sin P \sin Q)

= 2 + 2 (\cos (P + Q)) = 2 (1 + \cos (P + Q)) = 4 \cos^{2} \frac{P + Q}{2}

Now,

| ω | = \sqrt{\frac{N}{D}} = \tan \frac{P + Q}{2}

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