Question

Find the euler form for the complex number z = - 7 – 7i ?
1. \(7{e^{i\frac{{3\pi }}{4}}}\)
2. \(7{e^{ - i\frac{{3\pi }}{4}}}\)
3. \(7\sqrt 2 {e^{ - i\frac{{3\pi }}{4}}}\)
4. \(7\sqrt 2 {e^{i\frac{{3\pi }}{4}}}\)

Answer 1

Correct Answer - Option 3 : \(7\sqrt 2 {e^{ - i\frac{{3\pi }}{4}}}\)

CONCEPT:

A complex number z = x + iy can be expressed in the polar form z = re^iθ, where \(r = \sqrt {{x^2} + {y^2}} \) is its length and θ the angle between the vector and the horizontal axis. The fact x = r cos θ, y = r sin θ are consistent with Euler’s formula e^iθ = cos θ + isin θ.

CALCULATION:

Given complex number is z = - 7 – 7i

Here rcos θ = -7 and rsin θ = -7

By squaring and adding, we get –

\({{\rm{r}}^2}\left( {{\rm{co}}{{\rm{s}}^2}{\rm{\theta }} + {\rm{si}}{{\rm{n}}^2}{\rm{\theta }}} \right) = {\left( { - 7} \right)^2} + {\left( { - 7} \right)^2}\)

\(\therefore r = 7\sqrt 2 \)

\( \Rightarrow \cos\theta = \frac{{ - 7}}{r} = \frac{{ - 7}}{{7\sqrt 2 }} = - \frac{1}{{\sqrt 2 }}\) and \(\sin\theta = \frac{{ - 7}}{r} = \frac{{ - 7}}{{7\sqrt 2 }} = - \frac{1}{{\sqrt 2 }}\)

Since it is in third quadrant, \(\theta = - \pi + \frac{\pi }{4} = - \frac{{3\pi }}{4}\)

So, on comparing with z = r (cos θ + i sin θ), we can write as \(7\sqrt 2 \left( {\cos \left( { - \frac{{3\pi }}{4}} \right) + i\sin\left( { - \frac{{3\pi }}{4}} \right)} \right)\)

As we know e^iθ = cos θ + i sin θ.

So., the euler form of complex number could be written as \(z = r{e^{i\theta }}\)

\(\therefore z = - 7-7i = r{e^{i\theta }} = 7\sqrt 2 {e^{ - i\frac{{3\pi }}{4}}}\)