Write the complex number in the form a + bi. 8(cos 30° + i sin 30°)

Solution:

Given, 8(cos 30° + i sin 30°)

We have to write the complex form of the given term.

Using the Euler’s Formula,

\(re^{i\theta }=r(cos(\theta )+isin(\theta ))\) --- (1)

Where, \(a=rcos(\theta )\)

\(b=rsin(\theta )\)

So, \(tan(\theta )=\frac{b}{a}\)

From the given expression,

r = 8

\(\theta =30^{\circ}\)

Now, \(a=8(cos30^{\circ})\\=8(\frac{\sqrt{3}}{2})\\a=4\sqrt{3}\)

\(b=8(sin30^{\circ})\\=8(\frac{{1}}{2})\\b=4\)

Substituting the above values in (1)

r(cos(\theta )+isin(\theta ))\) = \(4\sqrt{3}+4i\)

Therefore, the complex number is \(4\sqrt{3}+4i\).

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Write the complex number in the form a + bi. 8(cos 30° + i sin 30°)

Summary:

The complex number in the form a + bi 8(cos 30° + i sin 30°) is \(4\sqrt{3}+4i\).