Question

The simplified form of expression \(\frac{{\cos 4\theta + i\sin4\theta }}{{\cos 5\theta + i\sin5\theta }}\) can be written as –
1. (cos 4θ + i sin 5θ)
2. (cos θ + i sin θ)
3. (cos 4θ - i sin5θ)
4. (cos θ - i sin θ)

Answer 1

Correct Answer - Option 4 : (cos θ - i sin θ)

CONCEPT:

According to Demoivre’s theorem \({\left( {\cos\theta + i\sin\theta } \right)^n} = (\cos \left( {n\theta } \right) + i\sin\left( {n\theta } \right))\)

We can write it in modulus argument form as [r, θ] ⁿ = [rⁿ, nθ]

CALCULATION:

Given expression is \(\frac{{\cos 4\theta + i\sin4\theta }}{{\cos 5\theta + i\sin5\theta }}\)

\( \Rightarrow \frac{{\cos 4\theta + i\sin4\theta }}{{\cos 5\theta + i\sin5\theta }} = \frac{{{{(\cos \theta + i\sin\theta )}^4}}}{{{{(\cos \theta + i\sin\theta )}^5}}}\)

\(\Rightarrow \frac{1}{{(\cos \theta + i\sin\theta )}} = {(\cos \theta + i\sin\theta )^{ - 1}}\)

⇒ (cos(-θ) + i sin(-θ) = (cos θ – i sin θ)