Question

Write each expression in the standard form for a complex number, a + bi.
a. [3(cos(27°)) + isin(27°)]5
b. [2(cos(40°)) + isin(40°)]6

I just really need help

Answer 1

Answer on Question #42778 – Math – Complex Analysis

Write each expression in the standard form for a complex number, $a + bi$.

a. $[3(\cos(27{}^\circ) + \sin(27{}^\circ))]5$

b. $[2(\cos(40{}^\circ) + \sin(40{}^\circ))]6$

Solution.

We will use the De Moivre's formula

$$(\cos x + i \sin x)^n = \cos nx + i \sin nx$$

That is,

$$\begin{array}{l} a. \left[ 3 (\cos (27{}^\circ) + i \sin (27{}^\circ)) \right]^5 = 3^5 \left[ (\cos (5 * 27{}^\circ) + i \sin (5 * 27{}^\circ)) \right] \\ = 3^5 (\cos (135{}^\circ) + i \sin (135{}^\circ)) = 243 * \left(- \frac{1}{\sqrt{2}}\right) + 243i * \frac{1}{\sqrt{2}}) = \frac{-243}{\sqrt{2}} + \frac{243}{\sqrt{2}}i \\ \end{array}$$ $$\begin{array}{l} b. \left[ 2 (\cos (40{}^\circ) + i \sin (40{}^\circ)) \right]^6 = 2^6 (\cos (240{}^\circ) + i \sin (240{}^\circ)) = 64 * \left(- \frac{1}{2}\right) + 64i \left(- \frac{\sqrt{3}}{2}\right) \\ = -32 - 32\sqrt{3}i \\ \end{array}$$

Answer. $a. \frac{-243}{\sqrt{2}} + \frac{243}{\sqrt{2}}i$ $b. -32 - 32\sqrt{3}i.$

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